DIsney strollers are not adjustable, nor do they have storage. The strollers at the Magic Kingdom are hard plastic with no padding or cushioning at all. If you want to get a look at the MK strollers, go to the Moms Panel home page. Near the top right there is a box that says "What is the Moms Panel?" In the box there is button that says "Watch our vacation tips." About two thirds of the way through
this video is a bit of footage of the Magic Kingdom strollers.
If you are at all thinking that your kids would nap in a stroller, I would strongly encourage you to bring your own from home. The Disney strollers are not very comfortable for sleeping. Another big reason to bring your own stroller is that the Disney strollers cannot leave the theme parks. At some resorts, the walk from the bus stop to your room could be ten minutes long. I always want a stroller for a trek like that, particularly if you have a diaper bag and packages.
The only time I really recommend that people rent the WDW strollers is if they have an older child (maybe 5 years old) who never really uses a stroller at home and might just get a bit tired from miles of theme park walking. Otherwise, bring your own.