I think most of the hotels at Downtown Disney have car rentals too. I know the Hilton does.
We cannot say enough good things about National Car Rental, so we always rent from them. We were in WDW on 9/11 - as you can imagine it was really stressful and chaotic and everyone was in a state of shock. We had rented a car from National. DH - who just wasn’t even thinking straight, went to the Car Care Center to explain that we wanted to keep the car and drive it up to North Dakota, where a friend would pick us up (no planes were flying of course). Well, he went in there and in his state of shock forgot the rental papers, his license, everything. After hearing our story, the lady at National said “just take the car - don’t worry about anything, just get home safe”. A couple of weeks after we were home we got a call from National asking us if we got home safely and refunding our money - for the entire time we had the car! THAT is customer service - we won’t rent from anyone else now.