I reckon the best campsite for walkable distance to the bars and restaurants is Camping Mer de Glace(http://www.chamonix-camping.com/uk/welcome.html). The other campsite suggested is in Argentiere by the looks of things. The web addresses are very similar. Camping mer de glace is very close to les Praz and is about 10-15 mins walk from the MBC bar or another 5-10 mins into the centre of chamonix. I’ve stayed there on several occasions. Even when busy there’s plenty of showers etc. I had to queue for 20 mins for a shower this past weekend at a campsite elsewhere in France.
Chamonix has opened the lifts to bikes already this season due to the lack of snow. I was up on the le tour just this past weekend. Be careful if you plan to use the trains to get access to further up the valley. They try and limit each train to 5 bikes which in the height of summer is a joke.
Enjoy!