The stab vests are used by the guards, after being unlocked in the bureaucracy menu, by the chief. All the guards (regular guards, armed guards and dog handlers) will wear the stab vests, if they are available. Once the stab vests are unlocked, all the guards will go to the armoury, and equip them. Each stab vest costs $200. The stab vest will give the guards an extra 50% protection, (they can take 50% more damage before being knocked out), but they will move 30% slower, but this disadvantage shouldn't stop you from equipping your guards with vests, as the stab vest is very useful, as the chances for a guard to die are smaller and also, the guards wearing stab vests will be more efficient in stopping a full-scale riot.
If you unlocked the stab vests, when you hire a new guard, he will cost $500, but he will also already wear the stab vest, so, you will have to pay an extra $200 for the vest.