We always take our own. Ds has been 4x between 13mos and 24mos, and he's been to DL 3x over the last year (he turned 3, 3mos ago). Now that he's bigger I would feel comfortable with him in a rental however he napped great in "his" stroller so I was never sorry we brought it. The rentals are hard plastic shells on wheels, except at MGM the seats are made of a meshlike fabric. For the most part, they don't look that comfortable to me. Plus at the end of the day, you can take your stroller with you past the gate. This is a bonus with older kids that are very sleepy and too tired to walk.
Check out the Stroller FAQ in the planning section at www.allearsnet.com for lots of info about strollers at WDW, including pictures of what the rentals look like. hth. And best of luck with the adoption!!!
We're taking 3 for this trip, we wouldn't rent, I hear nightmares about them costing too much, queues being huge and wasting most of the day, and them getting stolen outside attractions.
Take your own! Then you can get one which reclines and your LO will be comfortable!
By the way 2 of the strollers we are taking are ones we don't particularly like so we will probably hand them over to someone in the stroller queue on our last day. If they choose to throw them away then it's no great loss to us but it will have saved someone some money