In case there are any new or new-to-WDW nursing moms - here's a summary of my BF experience with a 3 month old and then a 4 month old (same kid, 2 trips):
In March and even April, it was not too hot to do the nursing tank under a T-shirt combo. I also had a couple of dresses that had a pull-up/aside panel, and these I wore with just nursing pads underneath, when it was hotter. I found they were the most discreet, since I don't use a cover, but pulling up the t-shirt and unlatching the nursing tank was easier.
It was not this dress, but had a front panel similar to it:
http://oldnavy.gap.com/browse/produ.../www.oldnavy.com/products/nursing-dresses.jsp
Anyway, we didn't use any of the baby centers; instead, I just tried to find quiet shaded benches or rides to nurse on. Many rides and shows are rather loud, and YDD is easily distracted, but we nursed on these rides:
MK: Small world, people mover, carousel of progress, Hall of Presidents, on benches and along walls near the Mad Tea Party, at the junction of Tomorrowland and Fantasyland, shady tables near Gaston's tavern
HS: Indiana Jones Show, in the shade near the exit to Tower of Terror, in the shade across from Rock n Roller Coaster
EP: (best for NIP!) - Spaceship Earth, the France movie, benches in China, benches in the play area after Test Track, and my favorite, a bench under a tree after you exit The Seas, it's way to the left and no one even looks in that direction when they leave the building.
AK: I'm sure there were good places to nurse, but the Lion King show was a little loud (though we made it work) and the shady area watching Everest a little crowded. We didn't spend too much time there.
DD also tended to get tired and want to nurse around dinner time, so we'd bring the carrier and I'd nurse her at dinner and then we'd put her in the boba and she'd fall asleep usually until we got back to the hotel and took her out and changed her for bed. Also she's slept through fireworks at MK, EP and HS - just got ear protection (baby Banz) and put her in the carrier and walked around.
Things I wouldn't recommend but ended up doing in a pinch:
I was never very good at nursing in a boba but I had to go to guest relations at MK to get a charge taken care of on our last night. Baby was fussing and screaming, I was wearing the old navy dress, so I nursed her, out of desperation, in the boba. After being done at guest relations around 8:30 pm and YDD was screaming again and not willing to nurse in the boba. I unhooked the boba, got her into cradle position, and walked out of MK and all the way back to BLT nursing. It was not easy and it wasn't all that pleasant, but with the baby quiet no one even noticed; it looked more like I was carrying a sleeping baby, and that was better than a screaming baby. That said, not something I will do again!
At BLT we had a full size fridge and a freezer, and I brought milk storage bags and a flat insulated bag, froze all the pumped milk and put it in checked baggage, it stayed frozen until we got home. This last trip at GF I didn't pump as much, but we had a super cold fridge in our room that froze my milk anyway.