I started the Haute Route in July a few years ago but couldn't finish. I started in Chamonix, moved to the haute route proper from there (turn left at the southern end of town) and everything went fine until I was hign in the mountains. Came to a Refuge to find that the pass ahead was thick with fresh snow, no one had got through for a week.
Like other people who soon arrived, we went throught a bit of dithering before deciding not to risk it and then hiked elsewhere. I ended up going towards Aosta, then reurned to Cormayeur, caught the bus through the tunnel to Chamonix and flew home.
I promised myself I would complete it but so far never have.
I had no problems camping on the Tour du Mont Blanc, Les Houche, Contamines, a free one (with restaurant) where the route turns sharp left towards Italy, Cormayurer, then a Refuge, then I think La Foully (the Dutch were everywhere!) Champex, La Forcaz, plus a few wild camps in between.