We took our 9 month old to WDW this past Dec and LOVED the baby care stations. We didn't use them too much for nursing, I am used to doing it any place and lost all modesty during child brith!!!!! They were very useful when you needed a quite place to get the kid to focus and relax.
MK, has one quite room with rocking chairs and a sign for "only nursing mothers." There was plugs in there, but it wasn't private, others mother can come in at any time.
Animal Kingdom has private rooms, with electraial plugs. That should be every good for it.
Epcot. I belive has private rooms also.
Never made it to MGM.
Keep in mind that when you enter the parks they are now looking into all bags and electrontics. Unknowing males may give you some strange looks and even a hard time. Try to get the female staff to inspect the pump bag.
Disney does a great job helping familes. Call guest services a head of time, they might even have pumps there.
Have a great trip!!!!!!!!