Chamonix has a wealth of places to stay, from hotels to several campsites. Click here for further details.
The other towns/villages on the route, Les Houches, Les Contamines, Courmayeur and Champex also have several choices.
The TMB is well-served by refuges and gites d'etapes:
Kev Reynold's guidebook provides telephone numbers for advance booking. You are unlikely to be turned away from a refuge, however you may find yourself underneath a table! Refuges are particularly busy in July and August.
Joining the Austrian Alpine Club will discount the price paid for most refuges by halving the overnight charges. Note that some refuges are privately owned (e.g. Rif. Walter Bonatti) and do not provide discounts.