What would be the better option for privacy. Logging into my windows partition and using zoom just for class logging right back out or install zoom on my linux partition?
If you're using X, it would be able to read your inputs for other applications and such, but if you don't do anything sensitive while it's running it still won't be able to do anything.
It could, so while you're using it you should make sure you don't have anything sensitive onscreen.
If your desktop supports Wayland at all, you could switch to it while using Zoom, even if other things don't work as well, then switch back when you aren't.