The weather will be your big issue. It can be amazing in October, sunny crisp perfect riding conditions, or it can be snowing to the valley floor. The last few years have been perfect autumns for biking, but that doesn’t meant this year will be.
For lifts, here in Chamonix the Brevent lift is open for the all saint’s holidays. so the last 2 weeks of October and 1st week of November, if you have a car (it’s easiest with a car) then you can combine that with the le Chable-ruinettes lift in Verbier (possibly weekends only by then) and the always open Dorenaz and Saleve lifts to easily get a week of lift accessed riding in.
Accommodation won’t be an issue, loads of airbnb available then. Town will be quiet as that’s when most of Chamonix goes on holiday but it won’t be a ghost town likemost of the alps.
Personally I’d go off to Sospel at that time of year though. Uplift with coolbus, selfguide from transprovence Ash. Sorted! http://ridesospel.com/trails/