Do I need to hurt zombies or kill them to level up my weapons?

I'm having a tricky time upgrading the Vesper in the Zombies gamemode, seeing as it's a hit-marker machine. I was wondering, do I just need to hurt zombies with the Vesper to get experience on the gun, or do I actually need to kill them? I would think I'd have to kill them, since I don't get player experience until I get a kill.

From my experience, only getting kills and assists is when you will see the progression of the weapon increase. This seems to hold true across the Call of Duty series, and the Wiki also states this in the Black Ops 3 section:

Weapons are leveled up through usage, such as getting kills or assists.

Though it says "usage," I've never once seen one of my weapons level up from just getting hit boxes. The Vesper is a rather weak gun as you've said. You can (as I'm sure you know) buff the weapon by pack-a-punching it. Once you reach level 11 with the gun, you can also equip FMJs for more damage.