If you want to use your own weight vest in the gym, then yep, you'd have to schlep it with you each time. If you're friendly with any of the trainers, or maybe the front desk crew, you might be able to ask them to store it for you someplace, but that's opening a can of worms (it can "walk off" mysteriously, people could use it when you need it, other trainers could use it with clients, etc).
I once had something similar happen with boxing pads when I was working as a trainer. I kept them to use with clients, but soon found out that they basically became "public property" regardless of where I stored them.
So yeah, best bet is to just bring it with you when you need it and bring it home after. It'll be a hassle, but, so's training.