Not gonna discuss the cheap options b/c your party is full of teens and young adults. Instead, I'm gonna encourage the grocery delivery option, and actually probably to choose this option once to twice a day to feed your group, getting enough food for 7 days in the beginning, and then doing it again, especially once you see what you really eat.
Sandwich fixings (to include meats, cheeses, tuna packets, PB/Nutella/honey - whatever your crew eats), rolls and bread, lettuce/tomato/onion/avocado (again, whatever you all eat), shelf-stable fruit, and chips/pretzels/cookies with condiments (mayo/mustard) would be on my list. I'd also have breakfast fixings - coffee for their machines, milk, tea, sweetener, juice (whatever you drink), cereal, granola bars, hard boiled eggs (also a salad fixing), donuts/muffins. And I'd buy water and alcohol for the drinkers. So many things you can eat cold and enjoy, especially in summer, and so many that don't need to see a fridge.
For shelf-stable fruit - apples, bananas, oranges/clementines, pears, (even peaches, but they can draw fruit flies).
Bring one good knife, one good cutting board, and some dish detergent, and then buy disposable plates, cups, bowls, silverware, paper towels, and ziploc bags. You will save a ton if you eat just 1-1.5 meals out/day on the trip and let your huge crew enjoy nice groceries for the other 1.5-2 meals...
Remember, a TS for 6 adults will run $200/meal at its cheapest and probably $400ish regularly (after tax/tip). And even every CS will likely be around $100/meal...