Munich and skiing the alps

The weather can be quite variable in the Alps at that time - snow is likely, but the low elevation ski areas will probably not be open. I have been to Garmish-Partenkirchen at that time and the small Christmas market in town is quite charming. The Zugspitze has skiing on the glacier at that time of the year, but the snow quality may be mixed depending on how cold it is. I brought my own boots and rented skis up on the mountain. You can rent boots up there too. It was snowing so hard one day that I took the train down to Innsbruck for the day and had a nice lunch and wander in their Christmas Market.

https://zugspitze.de/en/winter/skiarea/zugspitze

My favorite ski area in that area of Austria is Hochfugen and Hoch Zillertal reachable through Kaltenbach in the Zillertal just southeast of Innsbruck, but again at that time of year you will have to check on the local snow conditions when you are there.

https://www.zillertal.at/en.html

I usually stayed at Haus Monitzer in Kaltenbach when I have gone.
https://www.apartment-monitzer.at

Hi,
Your week is early skiing season - actually the first week for many resorts - so snow can be hit or miss. Therefore I wouldn't make any reservations, I would wait to see if there's snow in Garmisch (most convenient) and if not, head towards Innsbruck (Stubai has a glacier), Mayrofen,... Accomodation is plentiful.

Back when I was living in Germany (20 years ago), I was a member of the local German ski club. Usually we would ski in Tyrol starting in mid-November and Garmish in December/January. I think you'll be wise to check on the snow before making plans, but in some of the smaller towns you may not be able to get a room if you wait too late.

I'll recommend Hochgurgl and Obergurgl as excellent places to ski in Tyrol. Garmish, when the snow is good, is Olympic quality. I've never skied Innsbruck, but have spent a number of winter days wandering old town and have fond memories of those.

One place you might also consider is Slovakia. Many of the Germans and Austrians I know now go there for fun, and inexpensive, ski vacations.

Soelden, just below Obergurgl, has two glaciers connected by a tunnel you ski through, which is different. They have a World Cub Slalom event there in October, so they will have something. Big resorts have snow-making systems now, The economics are undeniable these days.