We went to Disney in October, and I had an exclusively breastfed 6 month old at the time. I am most definitely not a NIP type, but at Disney, I'd say it is a necessity to do it on occasion, especially if your baby won't take a bottle or a cup. I was usually able to find a shady spot, and I used a nursing cover most times. Sometimes, though, it was just too hot on Lauren. Of course, once you latch on, it's pretty discreet anyway and you don't really need a cover. Not once did I get a disapproving look or comment. Unfortunately, due to the heat, you will probably find yourself feeding more often. Take plenty of water into the parks just for you!
That being said, I did use the baby care centers at every park except Epcot. MK was the hardest, because the nursing room is not private and Lauren was very social and didn't want to eat while other people were there and talking. (She did fine outside, though. Go figure.
) Both AK and the Studios are fine.
If you have any specific questions, ask away!
My experience was that my nursing baby was much easier than my 3 year old and my 5 year old at all of the parks.
Kathy