Our first visit, my daughter had just turned 5 and we rented a double for her and my 8 year old nephew for a couple of days of a week long trip. She has always been really active and had little problems keeping up....most days. Epcot and Animal Kingdom were the ones that gave her problems, and even then it was later in the day that she got really tired.
My son is 5 this year, and he is a completely different animal. He isn't active, has low muscle tone, and never gets in a hurry, ever. I don't think he is anywhere near ready to go to Disney without a stroller.
So I guess my answer is, it depends on the kid. I would probably bring a stroller since the rental prices have gone up so much. Then if you don't need it, you can leave it in the room. Or, if you take it to the parks, you could park it in the stroller parking area that you will be closest to. If she gets tired, the stroller will be close by.