Hi, Adrienne!
When my kids were just little Mouseketeers, I liked bringing my own double stroller from home. I liked having it in the airport, at my resort, and when traveling to the buses or around Downtown Disney.
There are strollers for rent at the parks. Click
HERE for information and pricing. These might not be great for your 9 month old, as they are the hard plastic type that do not recline. I think you might be better off bringing your own cozy stroller from home that your kids are secure in and familiar with. The only downfall is that strollers need to be folded before boarding buses, but you will be an expert at that in no time!
Quick tip: when entering the parks, check for the Baby Care Centers which are great for feedings and changings. You will love these centers, I spent plenty of time there when my kids were in diapers!
Have a great trip with your kids and take lots of pictures to capture all the magical memories!