Early Vigilante level 12

Right, from what I hear people are still having problems with this, so I'm here to help.
Some of the tips I will mention here are things you may have heard already. So for those who know, yes mostly it is the old "use the motorbike" method, however, I will also give tips on how to reach your goal even when that isn't enough. So fear not I have a foolproof method to get you your improved body armour even if shooting from the back isn't enough.
With that said the motorbike method is good enough, only you probably were making certain mistakes, so I will also provide tips on how to do it properly.
But before we get to that I need to explain certain requirements you need to fulfil.

Important: The tips I provide here were only done in the original version, so I cannot confirm if all of them work in the remastered version. But anyone who plays the definitive edition is free to confirm if they do or don't work.

Technically, with the tips I'm going to provide, you can do this regardless of your story progress. So you can do it even while you have only done In the Beginning mission. However, at that point you will have no safehouse you can save your progress at, so you would need to earn some money first to purchase a safehouse.
You could do either the Taxi Driver or Paramedic side mission to earn some money.
However, it's best if you at least complete Big Smoke as that will give you access to the Johnson House, which will be helpful in more ways than just being a free safehouse, which I will explain soon.

Regardless you should complete the Paramedic mission as that will upgrade your healthbar.
Tip: The mission is much more easy to do at the Hospital in Montgomery as the patients will usually be closeby because it's a small rural town. Also there is less traffic in the rural areas so less chance to hit another vehicle and injure your patients.

Since you can't use the Rhino or Hunter at such an early stage the best alternative is the HPV-1000 police motorbike. It will allow you to shoot forward unlike any other police vehicle thus giving you a huge advantage against the criminals. However, it will not protect Carl like a Police Car would, so criminals may injure him directly. So you need to watch his health while persuing them, as criminals in the back seats can fire back at him.
Another reason why it's the best option, at this point in the game, is due to the fact it's much more easy to take shortcuts on a motorbike than using a car, which will come in very handy.

The most important question, as the weapons will decide which method will be used by you.
However, there are only 2 weapons you really need to consider, which are the SMG aka the MP5 submachine gun and the Minigun. You don't actually need both, depending on the tactic you choose you may just need one of them.

Getting an SMG and its ammo

A common mistake players make is using a Micro SMG aka the Uzi or the Tec-9 instead of the SMG. Which is understandable as it's much harder to get SMG's so early in the game. Also the SMG cannot be dual-wielded. However, the thing is that all submachine gun's share their ammo so you can build up ammo by collecting Micro SMG's and Tec-9's and then simply pick up an SMG which will now have all the ammo to itself. And while Micro SMG's and Tec-9's can be dual-wielded at Hitman level, you can only fire one while driving, so ultimately the SMG outshines them.

So how to get a lot of ammo? Well if you spray all the Tags you will get a free Tec-9 at the Johnson House which will give you 120 ammo and respawn each time you save and exit and re-enter the house. However, if you don't want to do that yet then there is another method: There is a Micro SMG (60 ammo) and Tec-9 (50 ammo) nearby the Johnson House, which will give you 110 ammo in total and respawn each time you enter the house and save the game.
Here is how you find them:

On the roof of Sweet's house to the left is the Tec-9. To get to it you need to first climb on the roof of the house to the right like so:

Now from there climb on the roof of Sweet's house:

And there is your Tec-9:

Next jump off the roof behind the fence and move to the right under the bridge where you will find the Micro SMG:

Then just get back into the Johnson House, save the game, then just rinse and repeat until you have enough ammo. 2000 - 3000 rounds shoud be more than enough. Once you have enough go grab the SMG which you can find in a corner of the area where the Lowrider Challenge takes place, right here:

Next train your SMG skill up to Hitman level, which shouldn't take long. If you need more ammo after reaching Hitman then just repeat the previous steps to get more, after which go grab the SMG again.

Getting a Minigun

Now this is a bit harder as there are no places in Los Santos where you can find one, so this will require you to go to Las Venturas to get it and earn an instant 4 star wanted level while at it.
But have no fear it's not as hard as you may think and I will guide you how to do it. The Minigun will have 200 rounds of ammo so you will need 3 visits to earn a full magazine, although how many visits you need to make will depend on the tactic you choose, so it may be more or less than that.
On the Screenshot below I have marked the location of the Minigun, while the character icon is pointing where you should enter Las Venturas from:

Once you are on land grab the nearest car you can find and head toward the location of the Minigun which is a construction site. Once you are close step out of the car and enter the location, here is the spot I use to enter it:

The Minigun is located on top of the building to the right. Good news is that while you are there the police won't bother you, although it's best to stay close to the estern fence, which will cause any police officers to spawn behind the fence where they can't do any harm. Now you need to climb the building using it's boardwalks. The first floor is easy but the following 2 will be hard and you will likely need a bit of practice before you get it right. To climb to the 2nd and 3rd floor you need to position CJ in a corner like so:

Then jump diagonally to the left to grab the upper boardwalk like this:

If your angle is too wide CJ will fall or if it's too thin he will hit the upper boarwalk with his head, so it's not easy but with practice you will eventually get it right. You can also do this by jumping diagonally right by standing on the other side of the corner if you prefer it that way. Once you are on the 3rd floor all that remains is to climb to the 4th floor which is easy as it's end is shorter than the 3rd floors boardwalk making it an easy climb, as you can see on this screenshot:

Once you are on the 4th floor the Minigun is in your reach, just run along the boardwalk to it's other end where you will find it:

Once you have the Minigun climb back down, you may take some damage while at it but nothing too severe. Now jump over the fence to the east and run like hell toward the water, no need to grab a car as it's a short distance. Then swim back to Los Santos under the eastern railtrack bridge which will provide some protection against police helicopters. By then your wanted level should drop to 3 stars. Now climb out of the water and get on the highway, grab the nearest car and run to the nearest Pay 'n' Spray or grab a police bribe to get down to 2 star wanted level, after that it may drop to 1 star on it's own and obviously after that it will drop to 0 very soon. Or just drive to the nearest safehouse and just save the game, which you have to do regardless.

Now, how many times you have to do this? Well depending on which tactic you choose even 1 visit may be enough. If you select the last tactic you will need at least 2 visits although it's best to make 3 then to be sure you don't run out of ammo. If you are a very poor shot or not good at saving ammo you can do more, however, more than 5 visits is pointless as that will be more than enough ammo even if you really suck at using the Minigun.

Oh yes the location also matters and it's best to do the Vigilante mission in the rural areas of Red County. The reason for this is because you will earn wanted level slower in rural areas. Probably because there are less witnesses and it's less likely to cause collateral damage. However, it's also better because you will likely earn more time each level, it's also more easy to take shortcuts in rural areas and there is less traffic to bother you.

Next is the rural town of Dillimore which you can use as a place of respite between levels if you need some:

As you can see on the other screenshot, to the left is a Vending Machine which you can use to restore your health, while to the right is a police station where you can find some Body Armour if you need it. As you probably noticed the town also has a Pay 'n' Spray you can use to lose your wanted level if it gets too high. Just grab the nearest vehicle, respray it and get back on your bike.

Note: There is no guarantee criminals will always spawn in the rural areas, some may also attempt to drive into Los Santos, at which point the next wave will likely spawn in the city. So it will be up to you to decide if you should continue the mission or retry. This also depends on the tactic you choose and depending on the level you are at already. The last tactic will not really require you to do the mission in rural areas at all, but it will make it more easy regardless.

Motorbike and submachine gun only

Some of you may have already tried this one, but if you are now using an SMG at Hitman skill level as I suggested it should be more easy. All you need to do is persue criminals and shoot at the back of their car until it explodes, preferably along with them. If they do manage to leave the car then you will have to shoot them or run over them. At higher levels they will carry more and more powerful weapons and their shots will become more accurate. Those that sit in the back seats may also shoot back at you if you get too close. At level 9+ some will use AK-47 to fire at you, which can kill CJ quickly, so if CJ is injured go to Dillimore to get your health back. You also need to be careful not to get too close to the car, as then the car may explode right in your face and destroy your HPV-1000. If you need to lose some heat then grab police bribes acattered around Red County or use the Pay 'n' Spray in Dillimore. With some practice you should be able to complete level 12.

Hybrid tactic

If you can't complete level 12 with just the SMG then this tactic will get you there. The tactic is pretty much the same except at level's 9-12 you will instead be using the Minigun. The Minigun will easily wipe out the criminals as it causes cars to explode in less than a second, so much so that it often causes a glitch where they explode twice, 1st due to the damage it cases and 2nd due to the burning engine. However, unlike the previous tactic it's best to corner the criminals, so you need to check where they are going and get in front of them. Then just get off your bike, aim at the cars as they are heading toward you and blow them to kingdom come. Less than 10 ammo is needed to blow up a vehicle, so 1-3 pick up's of the Minigun is enough.
As you can see on the screenshot I still had 505 ammo left when I completed level 12, yet when I started level 9 I had 600, so just 1 pick up of the Minigun would have been enough. However, with 2 or 3 you won't need to watch your ammo, so you won't need to be shy about using it in a more reckless manner.

Minigun only

Why use a submachine gun at all when you can just blow up the criminals fast with a Minigun? Rightly so, only for that you will have to make 2-5 pick up's of the Minigun, depending on your skill with it. This tactic goes pretty much the same as the Hybrid tactic at level's 9-12 only now you will be using the Minigun for all level's. Good news is that you don't need access to the Johnson House for this one. Also you won't really need to do this in the rural areas, as all you will need to worry about is your wanted level. However, it's more difficult to corner the criminals in the city as there are more turns they can take, so it's still better to use the rural areas for the mission. Unless you don't mind running after their cars each in turn and firing at them from the back.

As you can see I only used up 261 ammo out of the 1000 I started the mission with. So 2 visits would have been enough. In fact even 1 visit could be enough but that would be really pushing your luck, so unless you wish to make a challenge run out of it don't try it.

Well I guess that would be it, hope you found my guide helpful. If you still can't finish level 12 then I doubt there is a more simple tactic than this and you will just have to use a Rhino or Hunter once you have access to them. And before you ask, no a Rocket Launcher won't make it more easy, because the rockets are much slower so it's easy to miss your target with them, especially from a distance. In fact even if you land a hit it's very likely it won't blow up the vehicle instantly, so the criminals would just exit it and you would have to kill them individually. So chin up and I hope you get your improved Body Armour.
Cheers.