Hi Liz!
I'm so glad to hear you're planning a visit with a one year old! I can imagine the excitement that is building.
You should know the strollers available to rent in the Disney parks do not recline, as they consist of a hard plastic mold. Some parents have been able to make due with these, but I personally, have always prefered to bring my own stroller for use with my kids. If you are willing to give it a try, you can check out a few of the details regarding the costs of renting the strollers
here.
I've come to really appreciate the baby care centers inside the Disney parks. They are great for escaping the heat since they are nicely air conditioned and that makes for a great way to recharge for a bit. In addition, you'll find a changing area, a tv, a kitchen and sitting area (with high chairs, too), bathrooms, and private areas for nursing mothers. There are a few baby essential items for sale, too, should you need formula or diapers, but I find them to be a bit expensive so remember to bring your own. You can find the locations of each park's baby care center in any park map, but here's a rough idea of where you can find them:
Magic Kingdom: Near the Crystal Palace restaurant
Epcot: Future World's Odyssey Center
Hollywood Studios: Immediately to the left as your enter the park, in the Guest Services center.
Animal Kingdom: Discovery Island behind Creature Comforts
Hope this helps and that you have a magical vacation!