You can't be paid on an H4 visa. There used to be a possibility of getting an EAD while on an H4, but I believe that program has ended (and was pretty limited).
If this is all volunteer, it will be completely up to the institution. Some institutions require volunteers to go through onboarding (especially if you'll need access to medical records to do research). It should be legal for them to do so, but some may refuse. Same with clinical work.