A couple options:
Or if you really want some cheap Amazon crap, I bought this for a trip two years ago, and it's absolutely plastic. The interior has a softer material that feels a bit like ... low pile velvet? Flannel? It doesn't feel like it came from an animal to me. I don't think it would hold up well to the kind of heavy use a wallet gets, though.
And edit to add, if you're running out of card space: it'll depend on what kinds of cards you have, but if any of them are membership cards, just store them in your phone. I use StoCard, and I've never had a problem using a stored membership card in-store. If it's something like a gas card where you have to scan it at the pump, then maybe you can prioritize those for wallet space.