Yes I've been camping a few times with just the kids. My husband isn't a huge fan of camping so rather than him come and be unhappy I'd rather leave him at home. The first time we did it it was out of necessity. We'd been to a festival, it had chucked it down and we had to go camping again to dry the tent out as our garden has no grass, just a planted slope. DS was 2 and DD had turned 7 the week before.
It was a fantastic few days. As the first trip was so easy I bought a smaller tent (we only had a 6 man) and this year I've bought another 4 man tent which was 2nd hand but much better quality. It's a bigger pack size though so we'll have to see how that goes space wise.
We like small sites, without entertainment. We picked one which was 5 mintues walk from a beach, only had space for about 10 tent and 5 caravans, had a small toilet block and wash up area. There's also a van which does food at breakfast and tea time. It caters mostly for people going to or from the beach, but it's so handy when you don't want to cook. We budget for some takeaways. As it's small I don't have to worry about where the kids are as I can always see them. I hate big sites with playgrounds as I'd have to spend my time sat in a playground.
My tips are give the kids "very important" jobs which play to their interests. My DS usually has the important job of being "wind defeater!" he has to sit on the tent/footprint to stop it blowing away. He then becomes "Master of the Pegs" and is responsible for passing me pegs. It keeps him busy and close by. DD is pretty helpful at putting the tent up now that she's a bit taller. She quickly loses interest so them gets given the important job of prettifying. She sets all the bedding up and gets to play house.
Don't worry about bedtimes. Mine go to bed when it gets dark and they're exhausted and that's when I go to bed too. They fall asleep in minutes and it saves the arguments about not being tired. DD wakes early whatever happens but has learnt to quietly entertain herself while DS and I have a lie in.
Pitch close to the toilets. Mine are sensible enough to go to the loo without me, but I'm close enough to get there quickly if needed.
Make friends with your neighbours, especially if they have children. We made friends with a lovely family who had teenage kids who happily entertained my 2 for days (I kept offering to take mine away to give them a break but they really were happy to play with mine) meanwhile I sat round a BBQ with their parents drinking and watching the sunset. Another time my kids found a family who had a wood burner in their tent. The kids ended up watching a film in the warmth while us parents sat outside drinking and shivering slightly.
I love solo camping. It ends up being much more sociable. I hope you have a fab time. I've envious of the campervan :)