Hokay...
I'm not posting to offend, so please try to keep that in mind.
My wife is a certified lactation consultant as well as a breastfeeding mom (our older son until he was just over two years and our new addition--who's five months--currently), and I am an involved, observant, and empathetic father, so I feel like I have some perspective on this.
Insofar as WDW is concerned, a mother can breastfeed any time the child needs or desires it, anywhere she is allowed to be...in public or not. That's the law in Florida (and many other states as well, including California and Kansas, to name just two). If the thought of this disturbs you, I don't know what to say other than: "Don't look," and/or "Turn your head." After all, there's plenty of stuff to look at elsewhere---it IS Disney World. Should your children see a mother nursing in public, and have questions about it, then YOU will tell them something, whether it be that it's a mommy feeding her baby, or whatever "cover story" you might wish to spin (and I'm not saying that would be wrong; to each his or her own, especially where raising kids is concerned). I would imagine that no nursing mother, my wife included, would WANT to expose herself unnecessarily to passers-by or those in close proximity...but at the same time I understand that it COULD (and DOES) happen. And again, I would say, if that bothers you, then don't stare, avert your eyes, look anywhere else, and forget about it.
WDW has been kind enough to provide the baby care centers at each park, and having used them on numerous occasions (at DL, anyway, but I'm sure we'll hit them in FL on our upcoming trip), I can absolutely say they're very nice, cool, comfortable, and inviting environments for nursing or even just taking a break with the baby. However, I also agree wholeheartedly with many others in this thread who have correctly pointed out that they are very limited in their accessibility (as in, one per park), and this has by design ruled them out for the vast majority of infant feedings (especially in the early months). Sure, WDW could build more of them, and that would be a great thing, BUT Florida law has mandated that having such a place is not necessary, and therefore the baby center is an amenity only...nice, but not required.
I guess I'm basically just reiterating what others have said here, and maybe that's not such a bad thing. I will close by saying that my wife has always been quite modest, and that she will consistently try to be as discreet as possible when nursing in public. That being said, however, August or September in Florida is not the time to be having a receiving blanket (and yes, I know they're thin---but they're still BLANKETS) draped over your shoulder...much less your head. She does have a couple of nursing tops, but not enough for every day, and so there will almost certainly come a series of moments during our week long vacation that she will have to nurse the baby in public. All I can tell you is that she won't be intentionally trying to flash anyone, and if somebody should see something they disagree with I would hope they will have the decency and graciousness to...
Look away.

Hey, was that Goofy over there?!?!
Scott in MO