There is no international airport in Chamonix. The closest and most convenient is Geneva, Switzerland. (Due to a quirk of geography, Geneva is actually NORTHWEST of Chamonix…the same general direction as Paris.)
Geneva is about an hour from Chamonix by car, slightly longer by train. Most who come here use Geneva as their entry port, and there are many transportation options to get you from Geneva to the Mont Blanc area.
Another option is Paris. Though MUCH further from Chamonix (six hours by car; about the same or a little less by high speed and local trains) Paris is very often much cheaper to fly into. The cost of a longer train ticket must be factored in; but still, if you are flying in from North America, Paris may be a more affordable option than Geneva.
Note that transportation strikes in France are a very real possibility and can severely impact your plans.
Others we met flew in from the Italian cities of Turin (Torino) or Milan (Milano) and took a bus north; we have no details about this, but the distances are not small, as you can readily see from looking at a map.
We chose Paris as our entry point. Partly this was out of a desire to visit the City of Light and partly because there is a reasonably priced non-stop flight from RDU to Paris. We would not advise the Paris option unless you are going to be touring that city as well.
(We can provide very exacting details about the Paris option upon request.)
Check your passport and make sure that it is not expired prior to leaving. No visa is needed for a stay of less than 90 days in France, Italy or Switzerland, and there are no border crossings to worry about in the European Union. But make sure your paper work IS in order.
For those in the USA, The US State Department website has a lot of valuable information on travel rules, warning, notifications, etc.
The French Rail site is a also a good one for updates….
To book trains in France, we used the Trainline website. Very easy.
The best site on shuttle buses in the Chamonix area is the Chamonix.net, which has a lot of good travel links in general.