Hi Tiffany,
I have explored the magical Disney Parks while all three of my children were nursing so I can relate to your concern. You will find terrific
Baby Care Centers located in all Disney Theme Parks. These will be clearly labeled on the Map you pick up as you enter the park, or on the My Disney Experience application available for your smartphone. In these centers you will find private nursing rooms equipped with outlets, where you can nurse, or use a breast pump.
I remember at one time you could leave your breast pump with the first aid station in each park, but I spoke to a cast member who told me that was no longer the case. I have two suggestions on storing your pump. One option would be to store your pump in a locker rental located near the front of each park. The second option is one I have utilized, I have carried my pump with me and stored it in the bottom basket of our stoller, as it was similar in size to a purse or small bag. I would also make sure you bring a small cooler with ice packets so you can store your milk while you enjoy all the parks have to offer.
I hope you enjoy your visit with your little one. If you have any other questions or concerns about traveling with your infant, please let me know. I love to help fellow Disney-loving moms.
~Jen