The Car Care Center is right near the MK area. The car companies there are National and
Alamo.
Here is some info from the Disney website
"Alamo® Rent A Car, the Official Rental Car of Walt Disney World® Resort, is conveniently located at the Orlando International Airport and at the Car Care Center near the parking exit of the Magic Kingdom® Park. For Emerald Club customers, National Car Rental® is also located at the Car Care Center. Shuttle service to and from your location is provided seven days a week, Monday through Friday from 6:00 a.m. - 8:00 p.m. and Saturday and Sunday from 8:00 a.m. - 6:00 p.m. (Holiday hours may vary.) Shuttle service should be scheduled at least one hour in advance. Call (407) 824-3470 for shuttle service and reservations."
Book you car on line, when you are ready call to make an appointment for the shuttle. They will pick you up at your resort and take you over to the CCC. When you return the car they will take you back to your resort.
Dollar has a location over at Regal Sun Resort, I believe they have a shuttle service too.
There is also an Alamo car rental at the Dolphin Resort.
When we went to Sea World we rented a car for the day from the Car Care Center, it was easy and convenient. We liked having a car, this way we had more control over when we wanted to arrive and leave the park.