Random thoughts:
- I'd look into ordering groceries ahead (so you just pull up to the store, and the groceries come trotting out to you). OR have them delivered to your room; this is worth a couple dollars on vacation.
- I'd give up on the idea of bell services managing your groceries. With a whole week of park time ahead, I'd take time to go to the room /get set up first; otherwise you're just stretching your options too thin /risking your food being stored inappropriately, which could be an expensive problem.
- With limited refrigerator space, I'd consider getting a couple days' worth of groceries, then have a second delivery. You can plan /pay for the groceries from home so you don't waste any time ordering on vacation.
- You sound like you're driving? Bring a cooler. Or two. Use the refrigerator for cold cuts, etc., and keep things like drinks and cheese in the coolers.
- Something I learned about managing food on a cross-country driving trip: Keep a box for your bread, chips, etc. -- or they'll be squished. And bring a cutting board and sharp knife.
- Start saving up small mayo, mustard, etc. packets from fast food now.
- Consider that tuna and peanut butter won't go bad /don't need refrigeration. Jarred roasted red peppers are a good stand-in for tomatoes. Yogurt and bagged salads are also good choices. Likewise, canned ravioli is a decent quick meal.
- I wouldn't be satisfied with a "real meal" only every other day. Sandwiches are okay, but I'd get awfully tired of them. Consider kids' meals for everyone in the parks; they're relatively inexpensive. Pizza delivery or a bucket of chicken is easy to have delivered to the hotel room without breaking the budget. If you have a microwave, you can cook hot dogs or baked potatoes. Last thought for a cheap meal: go by the grocery's deli -- more expensive than sandwiches, but also more filling.