What you first need to establish is your temperatures. Body temp min and max; ambient min and max. Select the worst case. Then take some guesses as to how much heat you have to pump. Is the vest insulated on the outside or does it have to cool the ambient as well? I would blind guess that you will need about 200 watts of cooling by the time you figure your losses.
Peltier modules are inefficient, but have a much nicer form factor than an R-134A compressor (like what is sitting under your fridge). You will have to weigh the befits of each. With the low temperature differential it will likely turn out that Peltier modules are actually more efficient than a prebuilt compressor designed for a high temperature differential. Also keep in mind that if you pay top dollar you can get a lot more efficient peltier modules than the $5 Chinese ones. Here are some nice reasonably priced ones with good specifications; tetech.com. From the COP graph in the specifications pdf you can calculate very precisely how efficient a module will be for a given temperature differential.
Im not sure about your application, but remember that endothermic reactions are either not sustainable (throw them away when you are done) or require recharging. Recharging requires cooling; so you are just delaying the need to have a heat pump somewhere else in the process.