Hi Rachael!
I have explored both Walt Disney World and Disneyland while my little ones were nursing so I can relate to your concern. You will find well equipped
Baby Care Centers in each of the 4 theme parks. The Baby Care Centers will be clearly marked on the guide map you can pick up as you enter the park, or on the My Disney Experience application available for your smart phone. Inside the Baby Care Centers you will find nursing rooms equipped with outlets where you can nurse, or use a breast pump.
I remember quite a few years ago you were able to leave a breast pump at the First Aid Stations in the theme parks, but I have spoken to a Cast Member and this is no longer the case. I have a few ideas of how you can store your breast pump and nursing pillow while you explore the Most Magical Place on Earth. One option would be to store your nursing gear in a locker available near the front of each theme park. The second idea, is one I have utilized myself while touring. I have carried my breast pump and stored it in the bottom basket of our stroller, as it was similar in size to a purse or small diaper bag. I would also recommend you bring along some sort of cooling bag or pouch to keep your pumped milk cold. The First Aid Stations and any Quick Service Dining location can provide you ice if you need for the cooler. I have also attached an informative video I made about the Baby Care Centers for you to see the basic layout, and features of each.
Enjoy your upcoming visit with your little mouseketeer. If you have any other questions about traveling with infants please feel free to ask. I love to help out fellow Disney-loving moms whenever I have a chance.
Dream as Dreamers Do,
~Jen