It’s been a while since I’ve done the Chianti wine country thing, so don’t have specific recommendations on drivers. For that, do a search on TripAdvisor using one of the nearby larger towns as you noted and search under tours.
Also, in the event that you are not familiar with wineries in Tuscany, it’s worth noting that he area is nothing like Napa or other California wine destinations. For the most part, only larger wineries offer tours/tastings and most smaller ones require advanced reservations. These tend to be very relaxed (an hour or two) tastings with some finger food. The density of wineries in the area is much much lower (and further apart) than what we are accustomed to here in the US, so that will limit how many you can hit in a day. This may influence your decision about hiring a driver. Finally, almost all of these towns have communal tasting rooms where the smaller wineries are represented under one roof. Those experiences, though not on winery grounds, are much more accessible and no less authentic. Chin chin!