Not sure I understand whether you want to be picked up in Florence or in Rome!
If you're in Rome and want to be picked up and make your way to Tuscany and end up in Chianti, then the best solution is to look for a private guide that can either drive you or arrange for the drive. But starting from Rome, I sincerely cannot offer any personal recommendations as to who to contact.
If you were in Florence, I'd recommend contacting Elena Fulceri at
elena@florenceguide.net who is a great private tour guide and who can arrange the tour and the car with driver for you.