You can pack a mother load!!!
I went to the Dollar Store and Publix and this is what I was able to pack for a 5 day trip for 2 adults, 2 teens, and 1 BIG toddler:
First, make sure they have a microwave in your room and that you also will be packing in checked baggage!!!
1. Powdered drinks-Kool-aid or Crystal Lite; you can buy 3 or 4 bottles of water there and just refill from your tap or in the parks over and over again.
2. Granola bars and pop-tarts; you can use coupons and Buy 1 Get 1 sales to get these cheap-great for snacks or breakfast
3. Chef Boyardee microwavable cups; they make these in lots of flavors and they are great for a filling dinner and can be bought on sale and with coupons
4. Dried fruit and nut packs or mini cookie packs
IF YOU ARE RENTING A CAR-lots of coupons for dinner off site. Lots of people on the DIS Boards rent cars-maybe you could start a thread and share a ride with someone staying at your resort during the same time-pay the gas money or make the daddies babysit while the mommies make the run (even mommies need a break) so they can go together!
However, I would not pack meat or anything such as that. The money you save with the snacks above is gonna be great to begin with, but with a little planning, you will be okay. Fruit is some of the cheapest snacks you can buy in the park-add a granola bar and a mini-cookie pack and you have really given your younger kids everything they would want to eat...plus, if you buy the glow sticks and cheaper Disney toys at Wal-mart and such before you go (especially the necklaces which can be the first thing kids want), you can really save a fortune to really make eating at the parks a better option...
You can do it!!!

Just plan!