Mountain Biking and Bike Hire, Chamonix-Mont-Blanc
I live in Chamonix and you've got a couple of options available. There are three hire shops in town - Grand Bi, Legend and Zero G. Zero G is probably the one I'd recommend for mountain bikes; they are the most central and have a pretty decent Specialized bikes from this year. Their website is http://www.zerogchx.com.
Otherwise, you could consider bringing your own bike (if you have one!). Your local bike shop should give / sell you a cardboard bike box, and if you add enough packing and protection, the airline hopefully won't wreck it.