Yes they do have a shuttle, however it is not free. I believe that they contract with a local provider for service. You have to go to the Welcome Center by the pool to arrange the shuttle. Usually the day before you need it.
We are staying in a couple of weeks and, from what I've discovered in my research, it looks like it takes about 5-10 minutes to get to Animal Kingdom, 10-15 to get to EPCOT and Hollywood Studios, and 15-20 to get to Magic Kingdom. Not too shabby!
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