Look at your airline's website. What is their checked luggage limit??
Most airlines allow 2 50 lb containers per person of checked luggage.
With 5 people that would mean 10 large bags/footlockers and 500 lbs of camping gear/clothes.
You need at least 1 large tent to sleep in and one small tent for storage.
Air mattresses, a pump and bedding.
The folding bag chairs for each of you.
Some type of shelter to sit under outside your tent. (This can be a tarp with ropes/bungies tied to trees or a collapsible shelter).
clothes for each of you (I recommend only 3-4 days worth and wash)
a shower kit for each of you.
Shower shoes
40 ft Outdoor extension cord and a power strip
Depending on what time of year you are going, you will either need a fan or heated mattress pads.
A battery lantern and batteries.
For one of your checked containers I would recommend a plastic locking footlocker so that you can lock up valuable items (like cell phone chargers, camera chargers, extra memory chips, whatever). Anything really valuable, you might want to keep on your person, or keep at the outpost safe deposit boxes.
Your normal camping stuff won't be bothered.
You can use Magical Express and they will deliver all your stuff to your campsite. I don't know what time of day you are arriving, but try to get an arrival fairly early in the day so you have plenty of time to get set up.
Since you say this is a new airline, they probably don't participate in Resort Airline Checkin, so you will have to check all your stuff at the airport yourself. With 5 of you that shouldn't be a problem. Once you arrive on the lower level, if the skycaps aren't available immediately, one of your party can go in search of one, or one of your party can go in search of a couple of carts while the rest of the party waits with the luggage.
I agree with you that you shouldn't try to cook on top of flying and camping. I never do. (I don't cook at Ft. Wilderness when I drive down either.)
If you have strollers, gate check those and they won't count against your limits.