Hi, Lynn!
Great question!
There are both strollers for rent and strollers for purchase in the parks at Walt Disney World.
Walt Disney World stroller rentals are available just inside the entrance to each park, which also serves as the return site. Currently, single strollers (which are much more substantial, durable and rugged than the stroller in the link's photo) cost $15 a day, and double strollers are $31 a day. There are discounts available for "length of stay" rentals, and Guests need only pay once if hopping between two parks on any given day. Both of these strollers have a small storage pouch in the back and both come with retractable shades, too.
Guests may also purchase a sturdy Kolcraft umbrella-style stroller at certain shops in the parks which, as of April 2013, cost $54.95. This type includes a cup holder at the handle and a retracting sunshade as well.
When trying to decide if you'd like to rent or purchase, consider if you'll need the stroller once you
leave the park exit. We've often used our strollers to wheel sleeping little ones to a bus stop, boat dock or, if we're lucky, right into bed if we're staying in a resort that's adjacent to the monorail. In fact, when our little ones fell asleep on a resort bus, it was a blessing to have a stroller with us to help get that very heavy bundle from the resort bus stop to our room. If that won't be a concern of yours, or if you simply don't want to be bothered with carting the stroller onto a bus or into your car, then opt for the rental.
Either way, it's such a help when you're traveling to WDW with little ones! Thank God for strollers, right?
Send more questions as you think of them, Lynn; we love to help!