As a travel agent and Disney specialist, I am often asked by clients if they need a car at Walt Disney World. I always tell them that I am going to give them a very qualified "no." Whether you need a car depends on where you are staying, the length of your stay, and what you will be doing at Disney World and possibly in the Orlando area. If you are not staying at a Disney-owned and -operated resort inside Disney World (with the possible exception of the Swan and Dolphin resorts), you will absolutely need a car. If you are planning to eat or visit anything outside of the Disney Resort, you will need a car for at least part of your stay.
Some of the Disney hotels have better bus service than others. The largest ones--Pop Century, Port Orleans, and the Caribbean Beach--have the most frequent bus service. (I have left out Coronado Springs and the All-Star Resorts because they are just so far removed from most of the complex.) Generally, maybe with the exception of the Boardwalk Villas, any of the Disney resorts that is a part of the Disney Vacation Club has less-than-spectacular bus service. This is also true of the Wilderness Lodge and Ft. Wilderness.
I used to tell clients that my one rule was that I never use Downtown Disney as a transportation hub; I only take a bus there if I want to be there. This is no longer really possible. Downtown Disney is really the transportation hub now. The Transportation and Ticket Center (outside the Magic Kingdom) is really no longer the hub it was. Aside from the monorails that go to the Magic Kingdom, the three resorts on the monorail line, and the monorail going to EPCOT, there is not much going on here any longer. The only bus service is to Ft. Wilderness and the Wilderness Lodge, the three other major parks, and Shades of Green. (I am not focusing on the monorails here, because most travel on the Disney transportation system is done by bus, and some by water taxi).
There are tricks to using the Disney transportation system, and I publish information on that which I give to my clients. It can be a good system if you know how to use it properly. If not, you will spend most of your time waiting for buses or sitting on them. If you are staying at a Disney resort and are not renting a car, do yourself a favor and make sure your travel agent books your airport transfers on Disney's Magical Express. These are very nice coaches (nicer than the regular Mears transfers), and this service will be free at least until the end of next year.