Yes, it is possible. I've done it on two occasions, and plan to do it again this December. I usually stay 6 or 7 nights when I camp at Ft. Wilderness.
The airlines allows you two 50lb bags of checked luggage, one carryon, and one personal item each. (Check your airline website for maximum weight and dimensions)
As a solo person, I brought the following.
A large duffle bag
A rubbermaid action packer bin
A carryon-size rolling bag
A daypack
Packed in these, I had the following:
Two 8x8 dome tents (one for me to sleep in, one for my gear)
Two rain shelters/canopies
Folding Chair
Folding Stool
Twin Size Air Mattress
Electric Pump
Sheets, Pillow
Electric Blanket
Battery Lantern
Battery Tent Lites (2)
Battery Fan (2)
Toiletries
2 Towels
2 Washcloths
3 Changes of Clothing
Laundry Detergent
Dryer Sheets
Plastic Sheeting for Groundcloths
Stake Mallet
Laptop
Cell Phone
40 ft Extension Cord
Power Strip
Digital Camera
Docking Station
Alarm Clock
Extra Batteries
Quarter Rolls for Laundry
Shower Shoes
Garbage Bags
Small Collapsing Cooler (I went over to Wilderness Lodge and filled it with ice)
I don't cook at Ft. Wilderness either.
For two of you, you will have double the allowance I had, and will only have a few extra clothes, a queen-size air mattress, and an extra folding chair, so it should be easy.
I put anything that was small, but had weight to it in my carryon and daypack.
Lantern
Batteries
Tent Lites
Tent Fans
Extension Cord
Power Strip
Quarter Rolls
Stake Mallet
Electric Controller for Electric Blanket
Air Mattress Pump
Computer
Digital Camera
Chargers/Plugs for Camera/Computer
Large, and lightweight items went into my checked luggage.
I took the tents out of their bags to distribute the contents evenly between my checked luggage.
I set my tents up sideways facing each other and put a shelter over the entrances, then set up the second shelter in front of my tents and set up my chair and stool. If you don't mind sending me PM with your email, I can email you a picture of my campsite.
I was on commuter flights, so I knew that my rolling carry-on would be gate-checked, so didn't worry about overstuffing it to fit under the seat or in overheads. I don't know what type of flights you are taking, so if you are on larger planes, you need to take that into consideration.
One thing I'm going to do next time to save space and weight is to not bring my own towels. Ft. Wilderness has pool towels, so when I arrive, I'm going to grab a couple of those from the pool and use them for my showers, then return them at the end of the trip.
Hope this helps,
Carol