Hey there, Ellir!
Welcome to the Disney Parks Moms Panel. I can't wait for your trip to the Happiest Place on Earth.
I love that you already plan on utilizing the convenient
Baby Care Centers located in Disneyland Park and Disney California Adventure Park. These are great places to change and feed your little ones as well as privately nurse or pump. My toddler now also loves the tiny toilets they have here which are perfect for potty-training kiddos. Unfortunately storage of your pump or milk within the Baby Care Centers themselves is not permitted, but I do have some advice for you.
I suggest bringing a small cooler and your pump to the
locker rental areas located within each park or in the esplanade between the parks. You can bring your own ice pack or grab complimentary cups of ice from any of the Quick Service restaurants in either park. What I love about these lockers is that you can use them the whole day as many times as you'd like for one small daily fee. I have stored my pump, milk, and extra layers of clothing for chilly mornings or evenings. I also have peace of mind knowing my belongings are safe while we soak up the magic of the Disneyland Resort.
I hope this helps, and please return if you have any other questions as you plan your practically perfect trip. Have a Fantasmic day!
-Jennifer