The Vest, Gloves, Boots and Gear Perk system is not available in Zombies. Instead, receive a permanent boost (throughout a Mission, unless you fully die) to various in-game attributes by finding (or crafting) a Perk-a-Cola:
Deadshot Daiquiri
Aiming down sight moves to enemy critical location. Increase enemy critical damage.
Death Perception
Obscured enemies, chests, resources, and item drops are more easily spotted.
Elemental Pop
Every bullet you fire has a small chance to apply a random Ammo Mod effect.
Jugger-Nog
Increases maximum health.
PHD Flopper
Diving to prone triggers an explosion. The explosion increases the higher you fall. Immunity from fall damage while diving. Immunity from area-of-effect damage from weapons you are using.
Quick Revive
Reduce the health regeneration delay time by 50%. Reduce the time it takes to revive an ally by 50%.
Speed Cola
Drink to reload and replate armor faster.
Stamin-Up
Increases run and sprint speed.
Tombstone Soda
On death, you create a tombstone stash at that location, containing your backpack inventory in the next game.