I have never found anything on the 10th Mountain Division website referring to Ice Climbing but this is something that I would like to know as well. The other question would be how far are you willing to hike/ski from the hut to get to some decent climbing?
Last year I did the Eisman Hut North of Vail and right under the Gore Range...I could only assume that there has to be some ice somewhere up there that is worth getting onto. I would also have to assume that in the Southern Colorado hut system there is access to good climbing too.