How do I add my name to my dad's car title if he's in a facility and can't get to DMV?

Ask at the DMV. Every state will be different in what they require.

The answer depends on your state and if you are POA. In Maryland, you can barely do anything in person anymore. Just about everything is handled online. I have POA and was able to purchase Mom a car and register it with no problems at all. She had nothing to do with it, not even signing documents. Even without POA, you can probably have him sign everything at home and handle the in person stuff yourself.