The previous lights in my bathroom were "Hollywood" style. I'd like to replace it with a more traditional/modern vanity light. Question is, there doesn't appear to be an outlet/gang box like I'm used to. If I take this plate off, how would I mount the new fixture?
It doesn't appear that they used a junction box. It seems the romex is entering this fixture through that clamp on the right of that lamp socket. If you remove the several screws holding this plate to the wall, there is probably a hole in the drywall where the cable is. If there was a j-box it should have been behind those cutouts in the center.
So, if that's the case, you can buy an old work j-box, and cut and fit it yourself. Then mount your new fixture to that.
An old work j-box looks like this and mounts to the drywall. https://www.homedepot.com/p/Carlon-1-Gang-14-cu-in-Old-Work-PVC-Electrical-Outlet-Box-B114RB/1004040271