Actually, back in July 2003 I took my girls (ages 4 and 3) to MK for a Princess Day while my dh took our son (7) to Typhoon Lagoon. We were going to eat breakfast in the castle then do Fantasyland so I didn't bother with a stroller. Problem! Halfway down Main Street, dd(4) starts to complain about the heat and walking. Crying then ensues by the time we reach the castle. We were early for our PS so we rode Dumbo while we waited. Still crying and complaining. So, instead of getting upset at her, I went into Tinks and asked if I could rent a double stroller there or did I have to go back to the entrance. They said, No Problem! I filled out the paperwork and voila! the stroller was brought right to the register.
Now, they could have taken pity on me, but I do think you can rent them there IF they have any. This was by 8:30am in the morning. I don't know if they have them there all day.
Good Luck. I would give it a try if you need to.
Karen