This is not an FDA approved vaccine. You CAN NOT require someone to get an unapproved vaccine, That's why less then HALF of the US Military has gotten the vaccine. Normally, the DoD can order you to get it. They can't on an unapproved vaccine. It's also illegal to ask if someone has gotten the vaccine.
UVA will have to prove that no one is permitted on their campus without the vaccine. That means staff, janitors, city workers, visitors, tourists etc.
If you've had the virus, you don't need the vaccine. So why require it?
If you’ve had the virus more than 6 months ago, you do, in fact, need the vaccine.