Black Ops 3’s Zombies mode has its own separate XP progression system for the first time, similar to multiplayer. Therfore, weapons have to be earned by levelling up. Below are what weapons you unlock as you level up.
Submachine Guns
Vesper – Level 4
VMP – Level 8
Kuda – Level 16
Pharo – Level 23
Weevil – Level 29
Light Machine Guns
BRM – Level 12
Dingo – Level 19
48 Dredge – Level 27
Gorgon – Level 31
Assault Rifles
ICR-1 – Level 2
KN-44 – Level 7
M8A7 – Level 17
Sheiva – Level 20
HVK-30 – Level 25
Man-O-War – Level 31
Sniper Rifles
Locus – Level 11
Drakon – Level 14
SVG-100 – Level 24
Shotguns
KRM-262 – Level 5
Argus – Level 13
205 Brecci – Level 18
Haymaker 12 – Level 33
Secondaries
Bowie Knife – Unlocked from start
RK5 – Level 9
L-CAR 9 – Level 22
XM-53 – Level 28