Do NOT go in June, July or August!!
We went to Disney World in Orlando last June 18-22, 2010 and it was 100+ degrees every day! Even at nighttime it was very humid. There were people there with babies in strollers and harnesses, the babies just looked miserable. While the parents were busy chugging water, they didn't offer as much time to babies needs.
They also had them in just onesies because of the heat. I can only hope that they had tons of sunscreen on them because the FL sun is fast to give sunburns. My forearms burned while waiting in line for rides!
My vote would be during Spring Break, or even Halloween (heard from family that its fun that time of year), or Christmas.