wheelchair rentals close to Vatican

You might find this previous thread useful:

https://community.ricksteves.com/travel-forum/italy/rome-mobility-impaired

Have you done much reading up on managing Rome with a wheelchair? You will encounter some challenges, such as some restaurant bathrooms being down a flight of stairs in the basement, narrow sidewalks, cobblestones, attractions with areas that aren't accessible, etc. As you'll see if you do a search for threads on the subject (enter "Rome, Wheelchair" in the search box), people have done it, bless 'em, but it's best to be aware that things will be quite different than in the States, if that's where you're traveling from?

Is the person needing the chair able to walk or stand unassisted at all? And you've confirmed that the apartment building has a lift?

Doing a search for 'wheelchair rental, rome' this is one of many options that came up

http://www.mobilityequipmenthiredirect.com/mobility-equipment-hire/c447-wheelchair-hire-in-italy-rome/