Just a quick suggestion, because it sounds like you're going to be spending an awful lot of money for renting AND insurance on a stroller...
Go buy one from Wal-Mart. I don't mean the $15 el-cheapo. Go buy one for $40-$50. For that money, you can get a decent one. If you want a "fancy" one, go to Craigslist and find a fancy one that was $150 new and now sells for $50. Use it for your trip, then either sell it or if you don't want to hassle with that, donate it to another vacationing family at WDW. It'll cost far less and you won't have to worry about it getting stolen. Yes, stroller theft can happen there, but it's exceedingly rare...and when it does it's only the super high end strollers (which I think is foolish to bring to a park anyway). Nobody will touch your average run-of-the-mill stroller.