Susan:
Our family has taken our own stroller and rented a stroller--more on which option in just a second--but we've never had any trouble with folks taking our stroller or items in it. In fact, our trick to not only pick ours out from the crowd, but also ward off imaginary stroller swipers (and let me stress imaginary) is to tie one of those disposable diaper trash bags (you know the blue ones) to the stroller with a diaper and a small chocolate bar and then, as it gets warm, mush it to look like, well, an unappealing messy diaper! Who wants to have the stroller with the "dirty" diaper?!
I heartily recommend that you take your own stroller, especially if you're flying on a plane. You'll need it in the airports not to mention leaving the parks for the hotel at night and during the afternoon for naps. It will get no more wear and tear than if it had been at home and, as I mentioned before, we've never had trouble in the parks, so bring the cooler. In fact, with the money you save from
NOT renting a stroller at Disney you can buy yourself a new model and a new cooler when you get home!
Wishing y'all a magical vacation with little ones!
Whitney