My DD went to WDW for the first time at age 6 (it was such a great age to have her there!). It never even crossed my mind that she might need a stroller since she hadn't used one at home for a couple years and I didn't know about the DIS then. We went from sunup to sundown and beyond every day, and the kid was a trooper! She never even one time complained about being tired, even though she fell asleep standing on the bus one time!

She just woke up at our stop and walked back to the room.
In general, I think I was more exhausted at the end of the day than she was! This was in early October, so it was pretty hot, but not quite summer hot, FWIW. Funny, the older she gets, the MORE she complains about being tired. But since she's 9 now and we aren't going back to the stroller NOW, we just take breaks in the parks whenever anyone needs to rest.
Best of luck, OP! I know a lot of people like a stroller to also haul their stuff around, too, but I think it would be a PITA to try to keep track of one, and I don't carry anything around that can't be easily stored in my lightweight string backpack. And DD now has her own, to carry her autograph book and any souvies she buys in the parks.