In reply to shoulders: Nearest campsite with swimming I can think of is Les Contamines Montjoie. Camping le Pentet. Has an open air swimming lake and lots of other stuff for kids. About 40 mins from Chamonix.
The big lake at Passy 20 mins from Chamonix is a big attraction with all kind of activities for kids, swimming, boats, bike tracks, etc.
Also Samoens campsite has an open air lake. 50 mins from Chamonix.
There are lots of public / municipal open air pools in most of the mountain resorts, for example Les Carroz, two pools, slides, chips, etc.