Our first trip we had a 6 week old, a almost 2 year old, and an almost 3 year old.
We also traveled with friends who had a 8 week old, and almost 2 year old. So, yes, we had 5 kids under 3. It was FABULOUS!! We had a ball! The babies were great!
We both wore them almost the entire trip, wake up,feed baby, strap her in the sling, and it was great. My 6 week old rode everything (clearly without a height restriction), she didn't wake anyone up at night (we were even in a value).
It was december, we didn't worry about heat (although one day it was about 90). She ate when she wanted, we were nursing, so if she was hungry she ate (and on a few rides, haunted mansion, jungle boat, some shows, etc). The characters LOVED seeing the baby, they are not supposed to hold the baby, but mary poppins took her and told us to go enjoy our day! (kidding) And we have an amazing picture of mickey holding the baby, with my other two right next to him, its one of my all time favorite pictures from disney.
Honestly, it was a breeze, I felt fine. I would rather be at disney anyday, even 6 weeks after having a baby.
Disney does have the baby care centers, but we never used them. No need at that point (we did use them during another trip), but we just never found it necessary to use them.
Go, and have fun!