Lauren,
You are not required to have a cover up. Florida state law is very protective of nursing mothers. In fact, Florida has been a bit of a pioneer as far as mother's rights in this regard, being the first state to enact breastfeeding legislation back in 1993.
By law, you are allowed to feed your baby anywhere and you are exempt from all public indecency laws (as of course you should be!). If you'd like comprehensive information you can use
www.breastfeeding.org and search 'Florida law'.
I began researching this extensively after an unfortunate incident in my own state, which is not nearly as progressive as Florida and has only protected nursing mothers within the last two years. It was a great relief to me to know that I would be 'ok' in DIsney. And once I visited for the first time as a nursing mom, I realized that I was very far from alone. You will see
lots of breastfeeding mothers throughout the park, not just in the baby care centers.
I definitely believe a baby can be fed discreetly anywhere, and I have done it all over Disney. :) I found many dark rides and shows to be comfortable places to nurse, such as It's A Small World, the Hall of Presidents, Great Movie Ride, etc. So fear not and have a great vacation!