Are you planning on only renting one van? You could easily rent 1-2 vans at the Rome airport (FCO) - check www.autoeurope.com for rates.
Unless you are planning on spending time in Rome and then heading to Poppi, driving out of FCO is quite easy. It is an approximate 3.5 hour drive from Rome to Poppi (exiting off the A1 at Arezzo).
I am unsure if there are any car rental agencies located in Poppi or Bibbiena. The closest main offices for rentals just might be Arezzo.
So you could either: Train to Arezzo, pick up rental car there and drive to Poppi/Bibbiena. Avis is directly across the street from the train station and your best option.
or Rent at FCO and drive all the way. Many car rental options.
Make sure not to get one of those Ducato vans. They are like driving an ambulance! Difficult to park as well.
The problems with renting cars outside of major areas (airports, train stations) are the hours. Even in Arezzo, they are closed on Sundays. On Saturdays Avis closes at 4pm but Hertz (not near the train station) closes at 1pm. So pick up and returns need to be scheduled around this.
However, it often does not cost extra (or at least not much) to rent from one location in Italy and return at another. For example, pick it up in Rome and return in Arezzo or vice versa.
Make sure all drivers get an IDP - International Driver's Permit - at your local AAA office (membership not necessary). You will need a passport photo (they can take one for you as well). Cost is about $15 for the IDP. It is the law in Italy to have one.