We went in July with our not quite 4 month old. I'm a teacher so the times we can go are limited. We had a great trip.
I was not worried about much of anything in the time leading up to the trip except the heat and it turned out not to be a huge deal.
We got to the parks generally around 11 AM after she woke up and ate and left around 6-7. We only spent 3 days in the parks. 1 in Epcot and 2 in MK. I would not hesitate to do it again!
In the morning we put a little sunscreen on her legs and arms (I know it's not recommended until 6 months, but we picked the safest block we could find and I'd rather have her use sunblock than get a burn and pedi ok'd it) and she wore a hat and sunglasses. We kept her in the stroller when it was especially sunny and had a fan going on her. We mostly carried her around in a Beco and she did great!
There's a ton of places to get out of the heat and sun. We spent almost 2 hours in the aquarium in Epcot, riding Nemo and watching the fish.
Just make sure you offer to feed the baby more often and keep yourself hydrated if you're breastfeeding.
We used the baby care center in MK, but I also fed her on the TTC, in Philharmagic and Hall of Presidents.
As long as you're willing to take it at the baby's pace and not go commando, you'll be fine. I think we rode maybe 10 rides the whole time we were there, but we had a blast taking our daughter. She did enjoy it! She liked the characters, especially the princesses, and the parade.
I'd be happy to answer any questions.