I donated a car to Goodwill about 30 years ago. If you could drive it in, they would take it and allegedly use it to teach people to be mechanics.
I think if I were to do it now, I'd probably go with the local NPR station. They'll come and pick it up. Or, there's a charity around here called Vehicles for Change. They supposedly fix up the car and give it to a family that needs transportation. I haven't looked deeply into them, but if I had a car to donate, I'd research.