We just came back from Disney with my 3 year old and 15 month old. We have a double Maclaren. I love this stroller. The first time we took a single Maclaren and it was great too. If possible have something that reclines, for those afternoon snoozes. A cup holder and basket is a must. We left our stroller all the time, with diaper bag and gifts that we bought, and never had a problem. There are thousands of strollers everywhere, so everyone is in the same boat. The park rental strollers are big and roomy, but are made of hard plastic, so I'm not sure how comfortable that would be after a while, and it also looks uncomfy to sleep in. The good news with those, is that if it does disappear you can take your receipt the desk and get another.
Have a great time.