Here is a list of competitions organised in France by the French Federation of Alpine Clubs, these are by and large welcome to all skiers with a minimum of ski touring experience
http://www.ffcam.fr/index.php?&&alias=alpi-ski-tour-09
and this is the international calender with the French events organized by the FFME
http://www.ski-alpinisme.com/calendrier.php
Note that if you want to take part in competitions like the La Pierra Menta you need to get your entry in on the day the subscriptions open. To take part in the big events you would need to have 100,000 meters ski touring training over the season.
All events require the usual Avalanche Transceiver, Shovel, Probe as well Climbing Helmet and usually ice axe and crampons.