Coming from someone who lives in Florida, and must deal with the heat and humidity year round, make up is a must for me. Waterproof mascara is your friend. Also take advantage of primer. I used to hate it, and think it was pointless, but then my friend MADE me use it and I no longer have a problem with smudging/running make up.
At disney, I usually wear foundation and maybe some powder. Primer. Then instead of using either liquid eyeliner/pencil eyeliner, I take a darker shade of eyeshadow, wet it down, then use a thin small brush and apply it as liner. I find that it stays MUCH better. It also does not smudge as easily. Waterproof mascara. & thats it. The whole process takes less than 10 minutes for me, and I only re apply when we return to the hotel room for an afternoon break/get ready for a nice dinner.
Disney World is like WalMart. You think you're going to be the ugliest person in the place until you get there. And then seeing ugly people is inescapable. You're suddenly surrounded by people who wouldn't look pretty even if they put on make up. But they're having a great time. Don't worry about make up. Worry more about having a good time.