Hi Rebecca!
I have explored both Walt Disney World and Disneyland while my children were nursing so I can relate to your concern. You will find well equipped
Baby Care Centers in all Disney theme parks, these will be clearly marked on the guide map you can pick up as you enter, or on the
My Disney Experience application available for your smart phone. Inside the Baby Care Centers you will find private nursing rooms equipped with outlets where you can nurse, or use a breast pump.
I remember quite a few years ago you could 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 couple suggestions for storing your breast pump. One option would be to store your pump in a locker rental available near the front of each theme park. A second option is one I have utilized myself, I have carried my pump with me and stored it in the bottom basket of our stroller, as it was similar in size to a small diaper bag or purse. Please also bring a cooler bag so that you can store your milk and keep it cold. The First Aid stations and any Quick Service Disney Dining location can provide you with ice if you need it for your cooler.
I hope you have an amazing vacation with your little mouseketeer. If you have any other questions about traveling with infants, please let me know. I love to help out fellow Disney-loving moms.
Have a Magical Day,
~Jen