Some foods to consider:
Beef, lamb, all sorts of poultry.
Grains and starches:
Quinoa, Bulgur wheat, wheat, barley, couscous, potatoes, sweet potatoes, turnips
dry peas, lentils, dozens of kinds of dry beans
Instead of carrots, what about parsnips? Instead of green beans...heck, anyone can survive without green beans. But it looks like most greens (spinach, chard, kale, etc.), broccoli, cauliflower, all sorts of squash, celery, all sorts of peppers are not on your allergy list. It still leaves a good bit of variety.
Berries (straw, blue, black, rasp, cran, etc.), mango, papaya, apple, pineapple, and cactus pear all come quickly to mind.
Dairy substitutes: WIC gives goat's milk as an option, have you tried that? My DD did quite well with it. It's quite expensive, but if WIC will pay for it...
Also, consider that things that can be tolerated you may be fine putting them on a rotation diet for, rather than stress over total elimination.
For bread, you can make your own soda bread without using any of the allergens, since wheat isn't on the list. Biscuits, pancakes, potato pancakes, all are pretty easy and don't use yeast.
For thanksgiving, I would probably do something along these lines:
Turkey in oven
Crock pot 1: Sweet potatoes with brown sugar and pecans instead of marshmallows.
Broccoli cranberry salad is a yummy side, and cold. Any other side would do as well: pasta salad, perhaps? The ingredients listed on the pasta I buy is 100% durum wheat, nothing else.
I'd skip the green bean casserole. Make peas instead. Microwave or steam them on the stovetop.
Mashed potatoes w/ turkey gravy. Cook the potatoes the day before and reheat.
Energ-G egg replacer will work in pumpkin pie recipes, as will goat's milk. Homemade apple pie can be very simple with relatively few ingredients (apples, a little sugar, nutmeg). You could also make apple crisp with rolled wheat in place of oatmeal for the crunchies on top. Have you tried cooking with lard? If butter and coconut oil are out, it may be an option to check. Since it's pure fat the proteins that are likely allergens in pork are largely absent, and it's great for things like frying, and in pastry (i.e. pie crust). It's also very, very cheap.
The part about not knowing how to cook for a big family amuses me, honestly, because it's taken YEARS for me to learn how to NOT cook for way too many people, because I learned to cook in a family of six-seven people!
Mine are extensive. The kids lose a lot of their reaction before age 2, so we ease up a bit after that, but still try to limit.
In all of the kids combined there have been allergies to:
As you can see, it's very difficult. The cheese, rice and bread fillers aren't as easy in our house. We can only do those things once a week, max. The first 6 are the ones we avoid the most, as every child has reacted to them. The rest aren't as insidious and don't seem to cause much reaction in them anymore. I avoid it all. I'm currently able to eat beef, tomatoes, apples, squash, potatoes, enjoy life chocolate, and barley (with only salt as seasoning). Every new thing I've tried beyond those so far has been a bust. We do make my meals separately. The rest of the family could never survive on only those things. I couldn't even do it if it didn't directly effect my life (colic!). And just about everything we make is from scratch (always for me when nursing, but usually for the rest, too) which is probably why I hate cooking so much. No "semi-homemade" for me. Sigh...