Hi Claudia! We have used rental cars many times during our Walt Disney World vacations. We actually have a large motor coach that we bring for our fall stay but at times we do not tow a car and need to rent while at Disney.
We have found the best way to do this when you are a resort guest is to use the
Disney Car Care Center. You will find Alamo and National car rental companies and they always have a large inventory for you to choose from. When our large family joins us we need a large SUV or van and have never had difficulty getting those but strongly urge you to make advance reservations with them no matter what.
The best thing for you, since you say you will only need it for part of your stay, is that you can get complimentary shuttle service to and from your resort to the Car Care Center.
You can also find Alamo and National car rental services located in the Swan and Dolphin resorts near Epcot. If you are staying in one of the Boardwalk resorts this might be even more convenient.
You are sure to find cheaper rates through some of the airport companies but having the convenience of picking up and dropping off a car right at Walt Disney World is worth the few dollars you might save at the airport.