We do this every trip.
There is a Publix store very close to DTD, just go past DTD through the DTD hotels, I think its called Hotel Blvd. At the entrance turn right, go under the highway, next light turn left, follow the road past the outlet mall, Publix will be right past there. You can check out there sales ad for the time you are going to be there, and don't forget your coupons. This store has a great deli and bakery dept. and great wine selection.
Here is a few things that have worked for us.
Breakfast items -, cereal, milk, muffins, cereal bars, orange juice, donuts, bagels & cream cheese, pop tarts.
Drinks -- We bring a case of water, Gatorade, soda, juice / juice boxes & adult beverages, also the flavoring packets to add to water great for the parks.
Snacks - - granola bars, breakfast type bars ( watch the ones that have a lot of chocolate, can get messy quick)- Apples - will hold up the best during the day. I have a couple of plastic containers that we put chips, pretzels, anything that we don't want crushed, I picked them up at the dollar store, and we refill them for each day.
Pool Snacks - Here is where we really have saved money, Adult beverage at the pool bar are costly, so we bring are own to the pool, A soft sided cooler works great for this, I have a plastic container that has a freezer stick thingy that attaches to the lid and keeps the wine cold.

, NO Glass around the pool., throw in juice boxes or soda for the kids, We take deli meat and cheese rolled up, cheese and crackers, grapes, chips, carrot sticks w/ ranch ( for this buy the individual serving size of dressing)
In room - Sandwiches - deli meat and cheese - PB & J - for some reason our DD's would get back to the room and be like hungry wolves. This was always a way to fill them up with something familiar, and filling. Even when they were older, it began a tradition to make a PB & J sandwich once back at the room. Great times and memories.