We have stayed in Chamonix and around quite a few times, and here's some spots:
First of all: normally if you stay somewhere in the valley you get free passes for bus and train ;-)
Camping Les Mottets: not too far, Chamonix accessible by bus, bike or a long walk, not too expensive, quite quiet, and shady spots, nice view on glacier des Bossons, and a bakery van driving up every morning. There's a campsite just next to it, not stayed there before...
Camping Mer de Glace: somewhat more expensive, and seems to fill up quite a lot.
There's also a camp site very close to the center, but it didn't really appeal to us when we visited it.
Camping in Argentière: there's two, the nicest one we found is the one just before Grands Montets cable car station. Think the name of the camp site is Grands Montets but not sure. Argentière is quite a bit further up, but there's a train and a bus going to Chamonix. Argentière itself is also quite nice (hiking etc...) And there's two nice bakeries (the one direction Switzerland does nice breakfast with coffee etc...)
Camp site has super friendly owners, very relaxed atmosphere, they have a small bar on site. Camping spots are sunny. When it storms, it can get very windy. Normal toilets to be found
Other side of the mountain, but in La Thuile (Italy, close to col Petit St Bernard), there's a very cosy camp site as well. Super friendly lady, nice surroundings, quite a bit quieter than Chamonix, but also nice for walking around the mountains.
Also nice: Camp site in Beaufort. Beaufort is a bit more rustic, but with that also a lot quieter than Chamonix itself. There's a cheese makery, and lots of good mountain hikes.
All of the above have electricity.
If you need any other info on Chamonix (hiking/mountaineering/food/drinks/whatever) don't hesitate to drop me a message. I have spent quite some time there, doing all of the above, and might have some ideas ;-)
Grtz
Jan