Activities near Greve without a car

Without a car, I'd base myself in either Florence or Siena.

Ditto Roberto. The driving in this area is so easy, I cannot imagine trying to see anything without a car.
Could you clarify the other question? You are staying in Sorrento a week, correct? And then you want to tack on three or four days somewhere in the vicinity of Naples before heading to Florence?

I'm curious what agriturismo you'll be staying at. We stayed at Castello di Verrazzano for a week and loved it. The reason I'm asking is I can't recall an agriturismo within walking distance from the bus stop in Greve. Technically you could walk to the entrance of Verrazzano from either Greve or a bus stop nearby, but from the entrance to the accommodations is a hefty walk up the hill. I think it's nearly a mile to the winery from there.

Greve is a charming little town, but you can pretty much cover it in a morning. You could visit Radda or Castellina by bus or private car, but the bus would be inconvenient and a private car would be expensive. For instance, to drive to Castellina takes 18 minutes. To take the bus takes 2 hours. The alternative would be to get up in the morning and catch the bus to Florence and then take a tour from there. You'l probably have to take the 6:30 am bus and be sure to be back to catch the last bus which leaves Florence at 8:00pm.

Frankly, based our wonderful time with Verrazzano as a base, I can't imagine exploring the region without a car.

We will be staying in Sorrento for about 6-7 days. Afterwards we want to train to another location in Italy for 3-4 days. We have been thinking about taking the train from Sorrento to Florence and then bus or transfer to the Agriturismo Terre di Bacaccio. Terre di Bacaccio is a 13 minute walk to Greve according to Rome2rio.

I’ve driven several times in this area and enjoyed it. This time around we are hoping for a car free trip. Greve may be a tough home base. That’s the reason for reaching out. Any other ideas would be great.

You've already accomplished the holy grail of finding an agritusimo within walking distance, so I'm sure you could fill your time in your immediate surroundings for a few days, but it just seems like once you have paid for transfers to and from and for at least one day tour, you could have just rented a car (even if it mostly sits while you are in Greve). I'd only book a private driver if I wanted to taste wine somewhere outside of the vicinity.
It looks marvelous--now I am dreaming about a walk to Montefioralle!
There is plenty to do (see below), what in particular where you looking for?
https://www.chianti.com/greve-in-chianti/what-to-see.html
https://www.visittuscany.com/en/ideas/greve-in-chianti-6-must-sees/
https://lovefromtuscany.com/where-to-go/small-towns-in-tuscany/greve-in-chianti/
For touring options, I would ask your accommodation and contact the tourist office: https://www.visitchianti.net/where-to-go/municipalities/greve-in-chianti/?lang=en
They have a page with tours out of Greve: https://www.visittuscany.com/it/ricerca?query=&filters=%5B%7B%22values%22%3A%5B%22%2Fsystem%2Fcategories%2Fterritori%2Fchianti%2F%22%5D%2C%22key%22%3A%22territorio_related_it_lower%22%7D%5D&asset=offerta_turismo

bus or transfer to the Agriturismo Terre di Bacaccio. Terre di Bacaccio is a 13 minute walk to Greve according to Rome2rio.

Out of curiosity I checked and the good news is that there's a bus stop on 222 right in front of Agriturismo Terre di Bacaccio. It looks like a very short walk from the bus stop. You could certainly walk up to Greve easily. See here.

I'd look into the other bus stops along the 222. This may help. The bottom line is that is seems to be possible, but will take careful planning.

It's Baccio not Bacaccio.

Nights in Italy are very dark, and there are no sidewalks between the town and the driveway to the agriturismo, so be careful that they don't run you over at night. From the little chapel (where the driveway starts) up to the agriturismo, it's another 10-15 min walk up in total darkness, on the gravel driveway, so bring a flash light.

If you are going to Florence first, I don't know why you wouldn't want to rent a car and drive the 45 min to Greve. All the rental car companies are near the Florence station. Beats me, but to each his own.

Is it possible to get from Greve to Panzano easily without a car?