Are you aware that it is about 163 miles from Nice to Avignon? A private driver would not be cheap, and if it is not a round trip for you, he still has to go back empty. I concur that a train would be better. If you are looking for some sightseeing along the way, I would rent a car and stop when something caught my eye. If this is a day trip where the driver waits for you and drives you back to Nice, it would still be expensive and take a good 6 hours on the road (maybe more, if there is much traffic).
If you plan on staying in Avignon, I suggest taking the train there and then taking a wine tasting tour from Avignon. The tourist authority there would have a list of options for that.
Lots of times it is a big adjustment to plan drives in France, when we think of drives in the US. I live in Southern California and we don't hesitate to drive 60-70 miles to go to a restaurant, even if it takes 60-90 minutes to get there. But I would never hire a driver, even an Uber, for that long of a drive, even if I could find one who would do that. And Nice-Avignon is a minimum 3 hour trip, one way. I'm sure you could find someone if you paid the driver enough, but this is not like a limo drive around Napa and Sonoma Valleys.
Another thought...there are local vineyards just about everywhere in France., and the Riviera is no exception. Finding a driver or a private tour in that area should be easy, and you could get a look at the countryside as well.
Just some thoughts....best of luck in your planning!