Hello Laurel! Welcome to the Disney Parks Moms Panel.
I am getting a little weepy responding to your inquiry because the first time I took my kids to Walt Disney World Resort they were 4 and 2, as well. Needless to say, a solid stroller was essential to happy campers throughout our first 10-day experience where the Magic is Endless!
Double Stroller Rentals are available at all 4 Walt Disney World Resort Theme Parks and at Disney Springs. The daily rate for a double stroller is $31, or $27 per day, if you pay up front for multiple days. The Multiple-Day discount applies to using a double stroller at any of the 4 Walt Disney World Theme Parks and Disney Springs regardless of where you initially paid. By simply hanging on to your receipt and showing it to a Cast Member at your next Stroller Rental location, you will be provided a double stroller and be on your way.
Within all 4 Walt Disney World Theme Parks are
Baby Care Centers, which provide a complimentary location for feeding, rests or quiet time. The Baby Care Centers also sell various amenities to assist with any forgotten necessities, such as diapers, formula and over-the counter medications. They are wonderful spots to get away from the noise and excitement and take a breather with little ones (even a 4-year old).
Also, consider Memory Maker to take advantage of Professional Disney Photopass photographers
to get the best shots of your family's first visit. I have a picture of my family's first visit right in front of me when I type to remember that Magical First Visit!
Have a Magical Vacation!