Disney will transfer your luggage from one resort to another, and we've always found the process to be convenient and painless: we drop off our luggage at Bell Services before heading to a park (and make sure they now which resort we're moving to), spend the day in the park, and pick up our luggage when we arrive at our next resort in the evening. (Just understand that the luggage transfer process can take several hours: if you arrive at your second hotel before or shortly after regular check-in time, it's possible your luggage won't be there yet, so keep any essentials with you.)
If you're switching resorts with refrigerated items that need to stay cold, however, you'll have to transport them yourself. You have a couple of options:
1 - As Dave said above, Bell Services will still refrigerate items delivered by or for you (they just won't refrigerate things during/after luggage transfer when they're the ones doing the transferring). What my family does is to take a folding insulated reusable shopping bag or lunch bag with us (upon which I place a big luggage tag that says, "Keep Refrigerated," although that's probably overkill). On hotel switch day, we take a rideshare from one resort to the other, give all of our stuff to Bell Services to hold until our room is ready, and ensure that the insulated bag is placed in Bell Services' refrigerated storage.
2 - A second option -- especially when we'd rather spend our switch day in a park rather than schlepping our bags around -- is simply to use up all perishable food items before we transfer hotels. Then we simply buy new perishables at the next hotel (or via a second grocery order). On many occasions the only perishable foods we've had are a little milk and/or cream cheese, so it's less trouble to buy them in two "installments," one for each part of the split stay, than it is to buy larger containers to start and then try to keep them cold. As far as non-perishables, no matter who's doing the luggage transfer, we simply pack them all (even bottled water) into our rolling luggage to be transferred by Disney. If space is an issue, I have a large collapsible zip-top tote bag that I always bring to WDW to use as a "pool bag," and I press that into service to transfer any and all nonperishable foods that won't fit in our luggage.
