Not a bad idea, but if the goal is to keep people from using bad emulators, you can list banned emulators in the game's rules

Some games request that you specify the emu build in the run comment, so this isn't unheard of. I personally wouldn't be against it.

I really wouldn't mind doing this as a general rule of thumb, but is there an easy way to do with window capture in OBS, or do you pretty much just have to do the full screen capture?

I'd say in general capturing the full emulator window is good practice