It is possible but I would locate the taxi service ahead of time and not leave it to chance/luck.
I stayed at Quinta de la Rosa in 2020 and am returning this fall. Both times, I’ve asked the Quinta to recommend a driver and they put me in touch with someone.
Driving isn’t the best option IMO because the roads require paying a lot of attention, thus the driver doesn’t really get to see much. And, if you taste, you don’t drive - safely anyway.
There are many tours, especially day tours that are available. That’s what I did my first trip to the Douro and it gave me a good introduction without having to worry about transport or anything.