I have not arranged private transportation in Europe before. Recommendations for companies to contact? Thank you.
I would start by asking your hotel for recommendations. For 6 people plus luggage, you will need a full size (8 person) van.
Google: "NOLEGGIO CON CONDUCENTE VITERBO"
Then see what comes up
I would suggest you to contact a local tour company we used in the past
www.shoreexcursionsinitaly.net that in addition to shore excursions also provides tours from town to town.