I'm surprised nobody listed rides/places, so I'll do that for ya

. Keep in mind most nursing babies are too little to be scared or distracted.
Pirates: What's really great is it's long and dark. I requested to sit in the back row.
Haunted Mansion: Again, dark and slow.
Train: Great because you can nurse as long as you want and then ride to whatever stop suits you.
Tiki room, although my little guy found the LOUD thunderstorm to be a distraction
Whatever boat is operating, although less private.
Small world
The people mover was great (and you now know how old my "baby" is!)
Other good places to nurse:
Obviously the baby center
There are benches near where the tours meet that are often deserted between tour departures
There is a bench between tomorrowland and adventureland that is rather hidden and quiet
Given some time I could think of others that may or may not have mentioned.
DCA was not in existance when my son was nursing, but others may have good suggestions for that park.
BTW, I never used a "cover", just wore t-shirts and a nursing bra. I never felt anybody was uncomfortable, and I'm certain most people did not even know what he was doing. My theory is baby's need to be fed and breastmilk was made for babies. If breastfeeding bothers someone, then they should put a blanket over THEIR head

.
Best of luck and congrats on your new additon.