Hi Jessica,
I think you'll find parents on both sides of the bring your own or rent one debate. Personally I prefer bringing my own. The strollers that have for rent in the parks are hard plastic & not very comfortable - especially for smaller more active children that take naps in their strollers. You can go to
www.allears.net & search for strollers to see a picture of them. I like that my personal stroller has more storage space than the ones in the parks & I don't have to unload it to get back to my resort.
Depending on how long you are staying, it could be cheaper to bring your own. Strollers are available for rent in the parks for $15 per day or $13 per day for a Length of Stay Rental. They are available on a first come basis & can't be taken from the park in which they were rented. You can take your receipt to the next park on the same day & get another stroller (if available). If you think you'll want a stroller to get to & from the parks each day, then bringing you own is the way to go.
I've been bringing strollers to WDW for over 8 years now - many times during peak season when its super crowded & I've never had a problem with anyone taking my stroller. Here are a few stroller tips. Never leave any valuables in your stroller. There are designated parking areas for strollers at attractions. Don't panic if its not in the exact spot you left it as Cast Members rearrange the strollers for maximum use of the space. Tie a bright ribbon on your stroller to help you spot it more quickly when you return. Put some form of ID on your stroller. Bring a stroller fan, extra batteries & a rain cover. A white towel over the canopy helps reflect the heat in the summer months. Don't leave open food in your stroller - squirrels & birds are very bold when searching for food.
Hope you have an awesome trip filled with many magical memories :-)