I offered suggestions on a slightly different situation in Post #10, (Option #3 is the one most pertinent to your plans) in the thread http://www.flyertalk.com/forum/ameri...-aa-miles.html

Essentially you will need to get to San Pedro de Atacama and while Iquique (and Antofagasta) are on the coast, Calama is the airport you'd want as it is closest to San Pedro, albeit you can still make it to San Pedro from Iquique, but it'll take longer. Also remember, the trip into Bolivia is a multi-day trip but showcases the otherworldly Reserva Eduardo Avaroa on the way there and back.

Just do a Google search and it should return a bunch of operators you could take. Finally, be reminded that it is not an easy trip, and it is really quite a luck of the draw whether all goes well or you end up with a lousy driver/ vehicle. The risk of "bloqueo"s in Bolivia is ever present - these are road blocks erected as part of political or other demonstrations which completely shut down roads and access to towns. I planned the trip twice in the reverse direction (from Uyuni). The first time, a bloqueo prevented me from getting on the road at all from Uyuni. So I had to plan the whole thing a second time - and this time, everything was smooth (and the experience was obviously unforgettable, and cannot be replicated anywhere else in the world). All the major guidebooks explicate these warnings in great detail, so read up before you go and keep your ear to the ground.

None of this should make you even think of NOT doing it -- the only thing worse than getting stuck in Uyuni is not experiencing the trip at all, so by all means, plan, but keep a day or two as a buffer in case things get delayed. You don't want to miss your flight out of the area.