On our last two trips, staying at ASMu, I brought a small thermometer to monitor the beverage cooler temp and as far as I could see the temp never went above 40 degrees, which considered the "safe" range by the FDA. Our fruit, creamer, juice, etc., never felt tepid, it was always cold, but I guess I should say YMMV. If you are worried about this and are flying to Orlando, I'd check a hard-sided cooler. Pack it full of socks and underwear, and run some duct tape around it so your "unmentionables" don't end up on the conveyor belt at luggage claim, hahaha! This would extend your "refrigerator" space and if it's not an expensive cooler, you could leave it behind in the laundry room with a "free" sign on it, if you don't want to schlep it home.
For groceries, I'd do a delivery service to the room. Kroger doesn't have in-person stores in the Orlando area but they do delivery. We've used them 3 times now and the process has been seamless. They charge regular in-store prices and their delivery charge is cheap- and varies with the delivery time. Last trip, we had groceries delivered the night before check-in ($3 delivery fee) and held at bell services at WL until our room was ready. In addition to groceries, you could probably order paper plates/cups/bowls and plastic utensils so you don't have to schlep them.
If you are considering sandwich makings, I'd stick to ham and bologna, as these are highly processed and won't spoil if left unrefrigerated for several hours. However... did you know that thermoses keep foods cold as well as hot?
Amazon has individual ones (10oz?) for about $10 each. For a family of 4, getting one for each of you is cheaper than one QS lunch! This means you could microwave frozen/canned things (frozen dinners, spaghetti-os, etc) before leaving the hotel and you'd have a hot lunch (or make microwaved macaroni and cheese cups). However, because a thermos will keep food cold, too, this means you could make tuna or chicken salad (canned food, order mayo and keep in fridge)in the morning, take it to the parks in the thermos, and make your sandwiches at lunch time. Just bring along a hard-sided storage container so your bread doesn't get squooshed before lunch time (and I might recommend hamburger buns instead of bread... they seem sturdier to me). Same thing if you are bringing in chips or fruit; they'll get crushed in your park bag, so put 'em in a box.
Trader Joe's has "cold bags" for $7. These are insulated, soft-sided bags that have done a good job keeping my frozen foods solid on the 2 hour drive home from the closest TJs. You could fill zip-lock bags with ice to keep your perishables cool if you don't want to use a thermos, and then take meats or "salads" for lunch. You can pack everyone's lunch into this and check it into a locker as you enter a park. However... Amazon sells insulated sandwich bags if you want to make everyone's sandwiches in advance. They are foil, insulated sandwich bags, 5 for $15. I'd make the sandwiches without spreads or condiments, to keep the bread from getting soggy, and use the individual packets of ketchup, mayo, etc., that you can grab at the food court, to add when it's time for lunch.
I admire your budgeting strategies, but you also need to run the math. Buying a cooler, thermoses, groceries, etc. will add up (especially if you are buying individual frozen meals). Also consider the value of your time. You are gonna have to spend a fair amount of time each morning prepping whatever you take into the parks. I, personally, wouldn't want to add another hour on either end of my day, just to prep lunches. It's not that I like to "waste" money while at Disney; I'm just too damned tired at the end of a long park day to spend an additional hour prepping: Washing out dishes and lunch bags, doing multiple microwave runs, etc., is going to take time and energy on either end of the day. I don't have the energy or stamina for that. We keep our food budget in line by having kids meals at lunch time (we find it's enough food as we don't eat as much in the heat of FLorida) or split a full-sized meal. We don't buy drinks or desserts with our meals, either, but we do budget for ONE park snack each day (and I lean heavily on the "get something here you can't get anywhere else" strategy, like dole whips, mickey bars, frozen bananas, etc., so I'm not paying Disney prices for popcorn or chips). Everyone knows this before we ever go into a park, and although we don't give the kids spending money, I have a friend who told her kids that any "extra" snack comes out of their own pockets. She says it was amazing to see how little junk was consumed, once they were paying for it themselves!
