You are actually allowed to bring a cooler and food on board, subject to restrictions (size, types, etc).
I have multiple food allergies and always bring some food on board our
DCL cruises and we always end up bringing some of it back off the ship with us at the end of the cruise.
DCL's prohibited items list [subject to change]:
https://disneycruise.disney.go.com/faq/prohibited-items/list/
DCL's coolers and food policy:
https://disneycruise.disney.go.com/faq/prohibited-items/ice-chest-food-containers/
For food specifically, assuming you will only have non-perishable, commercially packaged, factory-sealed food: "Nonperishable snack foods in sealed, original packaging. All items must either be consumed prior to returning to your home port or remain unopened."
Since you would have more than just snacks for the cruise, probably a wise idea to reach out to DCL special services to ensure that DCL won't have an issue with it.
Here is the challenge: US Customs.
The foods you bring will have to be imported back into the US, so they are subject to US Customs rules and scrutiny. So they come up against the rules for things like fruits and vegetables and meats and dairy products -- and it isn't just "fresh" stuff, either. The US CBP website has information on importation of food.
All food it must be FULLY DECLARED to US Customs. Every little bit of it. BTW, this includes things like gum, candy, mints that you may have with you too. Expect that you will get to talk with an Agriculture officer, which could be a quick talk [all of ours have been <1 minute] or a longer one, so budget time in for it.
IF you have declared everything and they decide something is prohibited, then they seize it, but my understanding is you aren't fined *because* you declared it up front. OTOH, if you do NOT declare something and they find it, then you can be fined or more because you are smuggling, and there are extra fines if the item is on the prohibited list. So make sure to DECLARE ALL FOOD!!
We always take a sheet of paper and make an itemized list of every food item we have, including quantity. Then when we go through customs we have that list out with our declaration card. If they want to know what foods we have, we tell them we have an itemized list and show it to them. They may ask questions about specific items.
When we do this, we only bring foods that we have purchased in the USA or Canada. We **have the receipts with us** to prove where/when they were purchased [this is at the suggestion of US Customs when I contacted them and asked about bringing food back off our cruise]. And we make sure we know the food items are allowed per US rules.
You can always contact US CBP and ask them about the rules and about your specific situation and the specific types of foods you will have; I have found them quite helpful in the past.
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Some other options to consider:
We have done a DCL cruise a few different ways...
One is to spend a number of days at a WDW resort [and maybe or not visit the parks] before the cruise, and then ONE NIGHT at a WDW resort after the cruise. We use the DCL transfers to get us to/from the port. We really enjoy that last night at WDW: the last night and morning on the ship are kind of hectic - getting your luggage out the night before, the early morning on departure day, etc. Our bags are not packed the way we want them for flying, among other things. So it is nice to have one night at a WDW resort where we can relax, repack our bags, decompress a bit after the cruise, and prepare for the flight home. We do not do a park visit. This is all about enjoying the resort, possibly doing a "resort hop" and going to visit some other resorts [using Disney's internal transportation system] - e.g it can be fun to go do a "monorail resort hop" and visit the monorail resorts and Wilderness Lodge. Or go visit Animal Kingdom Lodge and see the animals ! We may schedule a table service meal at a WDW resort hotel, or just stick with QS meals.
If you go that route, then you can have bell services at your pre-stay hotel transfer a suitcase to your post-stay hotel and it will be waiting for you when you check in after your cruise.
Another option we have done, but it is more tiring ! (and you would need to confirm with the Hyatt that they will hold your bag) :
Pre-cruise stay at a WDW hotel but then for the last night, return to the airport and spend the night before the cruise at the Hyatt inside MCO airport. Now, this worked nicely while Disney had Magical Express, since the return trip to the airport was free, but in the future you'd have to budget for the return trip to the airport. We did this option because in the past the DCL transfer buses leaving the airport were the first ones to leave, starting around 9 am, which means they were the first to get to the port. So we lined up early for them [there are people waiting by 8 am] and managed to get out of the airport and to the port nice and early. Also, DCL has arrangements with the Hyatt so that your checked bags are picked up in the morning and taken to the ship, so you only have to bring yourself and your carryons to the bus [see threads elsewhere for info]. HOWEVER, things could change once they resume cruising, so you will have to confirm how things are working. If you go this route, then either use the bags storage place inside the airport listed in one of the previous posts, or, stay at the hyatt for a night post-cruise and have the hyatt store your bag [IF they will -- you would need to contact them and ask if they will do that]. Also, the hyatt offers day use rooms, something to consider if you will be leaving on a later flight the day of the end of your cruise.