We took DD last July and she turned 3 months old while we were there. I know, I know, but the trip was planned before I got pregant, we had borrowed our DVC points, etc.
Anyway, she did not have any problems. Didn't get hot, didn't get red faced, stayed hydrated. We used the baby care centers a good bit. It is just a cool place to take a break. If DS wanted to ride something, one of us would take him on the ride, while the other one pushed the stroller through an air-conditioned shop. Some attraction have stores attached, so that was a good place to wait. ToT by the photo view area has benches and is dark, cool and worked well for nursing.
We went back to the room to rest every afternoon. Just took sometime to cool off. We did everything we wanted to do, but didn't push is since we had been there before. It was not the ideal time, but we made the best of it. The crowds were a bigger problem than traveling with an infant.
We are headed back in June and she will be 13 months. My guess is it will be much more difficult than when we took her at 3 months.
Good luck!