How big is your car / boot?
Look at the 5 person tents in the Millers sale. A tent with fibreglass poles will pack up smaller than steel poles or air beam.
Or go to GoOutdoors it big camping shop and look at pack sizes.
If you buy a tent with a small living area get a tarp to cook under if it is wet.
A folding chair each, a table, maybe one of the ones with a roll up top, a small low table to put a one ring stove on.
A roll up foam mat / SIM for each person, sleeping bag for each person.
‘Kitchen Box ‘ : don’t go mad: plan meals that can be cooked either either in one frying pan or one saucepan. (Take a ready made chilli, curry, stew etc). Have a look at the ‘collaps’ Stuff, a fold flat casserole pot/ kettle/ washing up bowl.
We go to ‘wild’ style sites and cook on a campfire so I don’t take a BBQ, I have one iron casserole that can go in the embers and in which I can cook a while chicken surrounded by spuds and carrots, and a grill that folds flat for barbecuing or putting a pot on. This means no gas bottle, big stove, BBQ etc..
Depending on your car and how the kids sit you have a spare seat space. You can fit loads of stuff under seats, footwells, under the kids feet. Soft stuff like sleeping bags on their laps, mats standing up next to their legs.
My cool box sits in the middle seat with the seat belt round it, or jammed in the footwell of the empty passenger seat (seat jammed up against it).
It’s big rigid things like tables that are awkward to pack. A small table to cook and prepare food on (mine is roll top) and they can eat if laps or a picnic rug if you maybe stick to non sloppy food like burgers, hit dogs, quesadillas (in frying pan) , risotto. I also use enamel ‘rice plates ‘ which have higher sides, like v shallow bowls, so kids spill less from plates.