When deciding which airport to fly to for Avoriaz, in this instance we have only one main recommendation: Geneva airport.
Geneva is one of the busiest International airports giving good access to most of the nearby ski resorts and summer destinations, year round. Other airports such as Lyon, Grenoble and Chambery are all options but as we'll mention below, these other airports are fraught with transfer complications and costs which are much more easily avoided if you opt for Geneva. One workaround to transfer issues is car hire, if you're comfortable driving in inclement conditions, in winter at least.
If you're wondering why we haven't covered Turin, Milan, or any other airports, it's because they just aren't recommended for flying to the French ski resorts... If you do end up there, you'll need to self-drive the rest of the way (and don't overlook some of the mountain-passes shown on apps/route-planners are shut in winter as they're snowed up, or will be closed due to snow!).
Some resorts, such as Courchevel, also have their own airports capable of taking reasonably sized private planes and helicopters.