Does Rick Steves recommend a specific tour company to see all of the Vatican?
You have to be the Pope to see "all of the Vatican," and Leo, if you're reading this, you have access to better guides than anyone on here.
For tours of the "public" areas (Musuems and Basilica) Walks of Italy, Through Eternity, Liv or the Roman Guy are commonly recommended companies. Make sure your tour specifically includes the shortcut between the Vatican Museums and St. Peters if you're interested in that.
There is the "Catholic Traveler" and have you checked the Rick Steves Guidebook for local guides?