Hi Keri,
Congratulations on your upcoming trip! There are many suggestions on the panel regarding strollers, so feel free to search the word for a variety of opinions!
Regarding whether to rent or bring a stroller, my vote is to bring your own for a number of reasons. First, to rent a double at Disney you'll pay $31 per day currently. (If you get a length of stay rental, you'll save a few dollars). For the cost of a double rental, you can purchase two relatively high-end umbrella strollers with good-sized canopies, storage baskets and even a few other "bells and whistles." I picked up two gently used MacClaren strollers on eBay that work great for our Disney trips for less than the cost of a renting a double for a week. Next, it's very convenient to have access to a stroller outside the parks, where strollers aren't available for rent. The airport, the Boardwalk at night, and your resort are prime examples of where a set of wheels will come in handy. Disney has what I believe to be a magical power to lull children to sleep just as their tired parents are approaching the stroller rental return. The prospect of carrying one or two kids out of the parks toward the bus stops or parking lots after a long day in the parks is not my idea of fun! Finally, the rentals don't come with too many bells and whistles (like a storage basket or sturdy cup holders) that come in handy during a long day in the parks.
Whatever set of wheels you pick, have a great trip!
Tanya