Each park has Baby Care Centers. Magic Kingdom and Epcot each have a room with several rocking chairs and you will not be alone, it was very nice. MGM has little individual rooms, like a dressing room almost for BF'ing. I did not use the one in AK so no help there. They also have changing areas. MK & Epc also have a tv area that has Disney movies, of course, playing if you have an older child. DD5 enjoyed this.
We went last Sept. with DS8, DD5 & DD3 mo. I am not one to nurse in public but did manage to in some places. I bought a nursing cover @ Target. I rode all the rides that did not have a height restriction. DH would ask for the back row on rides like Small World, Pirates of the Carib, Great Movie Ride, can't think of them all. I would nurse DD on the back row and during some of the shows. Nobody paid any attention except one lady commented on my nursing cover and wanted one. We did great. Have a great trip.