I did this about five or six years ago in Austria (Carinthia) - awesome experience.
Where I was there were loads of chamois - in the National Park areas (Hohe Tuern) I would say a gross over population. I went at the end of October and the first snow had just fallen - the weather was very variable - tea shirts one day full kit the next.
Being Austria the system is very strict - the quotas are absolute and the guides have to shoot exactly the sex and age class allocated to them. You can go through an agent and pay accordingly - I did'nt and paid for the flights, car hire, guest house, guide and trophy fee seperately. Not as taxing as it sounds but the system itself is complicated - the trophy belongs to the commune through the game office, the carcase belongs to the landowner and you must employ an approved local guide (e50/day). You need a contact though. At the time the trophy fee was about e1100.
The season is from about August until November or December - when the snow starts the chamois slowly move down into the trees but its only the depth of the snow that stops them , they are easily capable of coping with the cold and wind. Earlier inthe season you can get to the tree line by car but later you will need to climb - with shorter days this really shortens the hunting and the first time I was there we actually stayed in a hut just on the tree line. The winter coats later inthe season are much prettier.