If your kids can sleep sitting straight up on hard plastic (not to mention over the age of 2), they can sleep in the Disney strollers. Otherwise, I'd bring your own. It's highly unlikely anything will happen to your stroller, just mark it in some unique way (tie a scarf to the handle) as surely someone else might have your stroller, or one similiar. And that is just to help you find it faster.
How difficult it is to rent will depend on what time of year you are going and at what time you are trying to get a stroller.. You can rent for multiple days at one time, speeding up the process, but you will have to go through the process each day. Several times if you park hop. You can forgo returning the stroller if you don't care about getting your deposit back. Renting a stroller every day can get expensive, even with the multi-day rental price break (10%?). In which case, you may want to to just buy a less expensive stroller for the trip. Or rent from a baby item rental place. I think there are a few different links on the disboards. Plus there is no storage on the rental strollers.
Rentals are for the parks only. Walking to and from hotels/trams/cars, you are on your own. Depending on your child(ren)'s age(s) & size, you might want to consider that. And if flying, either way, do you want to have a stroller with you? Some people don't want to drag a stroller through an airport, others don't want to end up without it.