Yes, unless they have a very bad winter, they should be operating at that time. I was there a few years ago, for a week starting Dec 31, and they were skiing. The biggest ski mountain in the area, the Nebelhorn is only about 7000 ft at the top, but the town of Oberstdorf is at just over 2000 ft, so you get almost 5000 ft of vertical, more, I think, than any ski resort in the US. The town of Oberstdorf naturally looks like what Vail tries to look like. There is a Therme (what the Germans call a Spa) in Oberstdorf as well as one in nearby Sonthofen.
See the websites for Oberstdorf and Kleinwalsertal.