I was there last month with my 5 yr old son & if I had not had a stroller, it would have been terrible! Their little legs just can't keep up that pace and since most fireworks/night shows are over past his normal bedtime, he was tired! I also liked the fact that when it was busy during parade times/exiting the park at night, he was sitting safe, right in front of me.
We fly down from Indianapolis and have never had a problem gate-checking a stroller. Check your bags, take it with you thru security, be sure to get a tag from your airlines counter when you're waiting to board. The person that takes your boarding passes will tell you where to leave the stroller. Then it's waiting for you after you land as you exit the plane.
I also found several umbrella type strollers on
Walmart's site for around $40. The one I bought was by First Years and holds up to 50 lbs. It folded up easy to get on the bus back to our resort, and helped to have it again at the resort for walking back to our room from the bus stop at the end of the night. Also cheaper than renting if you're there more than 3 days.