Traveling from Rome to Montepulciano

The trains run on Sunday. A train Rome to Siena is not direct requires a change in Florence. Maybe take the train Rome direct to Chiusi-chianciano terme and rent the car there and drive to Montepulciano. Or take the bus direct Rome to Siena and rent the car there; or rent the car in Rome and drive to Montepulciano.

Trains travel even on Christmas Day.
In your case I would rent a car in Rome and drive from the start. Why go through the hassle of taking the train?
One more thing. Renting a car on Sunday is possible only at Rome Termini train station or Via Sardegna (Europcar), or via Veneto (Hertz) or at the airports. All other locations (including Siena or Chiusi) will be closed.

Renting a car on Sunday is possible only at Rome Termini train station or at the airports.

Not true. Some car rental offices are open on Sunday in Rome and elsewhere, usually with limited hours, like 8:30 to 12:30 or 1 PM. Visit your car rental website and search for local information to find out more details.

We picked up a rental car on a Sunday near ViaVeneto in Rome.

Almost all of the Via Veneto/Via Sardegna rental offices in Rome are open on Sunday.

I added the city center locations of Via Sardegna (Europcar) and Via Veneto (Hertz), which are open on Sunday morning. Actually I could have added also the 2 IKEA locations, which are also open on Sunday, but they are far from the city center.
The point is that on Sunday very few locations are open through Italy. Basically only at the central station of major cities and the airports.
For example Hertz has 34 locations in Rome, but only 6 (2 at the two airports, 1 at Termini station, 1 at Via Veneto, 2 at IKEA) are open on Sunday. If you take the train to Orvieto, or Chiusi, or Siena, or whatever, you won’t be able to get your rental car. Once you head north from Rome, the next one open location is Florence downtown (morning) or Florence Airport.

Budget, Avis, and Locauto are also open on Sunday (until 1 PM) at the Via Veneto/Via Sardegna location.

I'd pick up at the airport which will be possible to do on a Sunday. The stretch between Rome and Montepulciano is glorious -- so many wonderful towns to visit along the way. It is beautiful country. We stopped in Sovana, got a local map of the Etruscan tomb sites in the hills nearby and visited some in situ Etruscan tombs. We stopped at Saturnia for the amazing free public park hot springs waterfall; I smelled like sulphur through several showers but it was worth it. Getting from Rome to Siena is very awkward - no direct trains. The drive is wonderful. You could probably pick up a car on Sunday at Termini as well.