Hi Kristie!
I have breastfed both my kids at Disney World. It is easy and comfortable if you go to the Baby Care Centers. First of all they are AIR CONDITIONED! Each Baby Care Center has diaper changing areas, nursing areas, high chairs and even a TV for the older kids to watch.
I've been to the Baby Care centers at Epcot, The Animal Kingdom and Magic Kingdom. The Animal Kingdom is as nice as being at home--private nursing rooms with a glider and a changing table. Epcot has private curtained areas for nursing. While Magic Kingdom just has a small semi-private nursing room filled with rocking chairs.
I did not specifically notice an outlet for a pump, but I'm sure they can accommodate that, they have everything else--including many baby supplies (diapers, formula, etc.). Plus, the cast members at the Baby Care centers are always helpful (and quick to ooh and ahh over my cute kids!)
I also suggest getting a nursing shawl/cover. It helps when you have to nurse in line or while waiting for Fantasmic! :)
(Here's a link for the location of all the Baby Care Centers. And make sure you check out your park map for the little pacifier icon when you get there, too!)