I don't think there's any real difference, except that "stand a chance of" is something I'd be much more likely to say. I was going to say that "stand a chance at" is probably British English, but Gary's comment implies that it isn't, at least not in New Zealand.

chance at" = "chance of succeeding at" in this case. The version with "at" also sounded odd to me initially. "of" links the general noun "chance" to the information which is needed to define or specifies it. "at" doesn't have this function and chance is too general a word to have a clear meaning without other defining words. However, I then looked up "a chance at" and "stand a chance at" in a sentence database and found lots of examples, all from American sources. See http://fraze.it/n_search.jsp? and hardm=1&t=0&l=0&p=4&q=chance+at and http://fraze.it/n_search.jsp?q=stand+a+chance+at&l=0&t=0&ffo=false&findid=-1&ff= Based on this, I think we would have to say that "stand a chance at sth" is acceptable at least in informal American English as a reduced form of "stand a chance of succeeding at sth".

I'm not sure. To me, the second sounds wrong, and is something that I would never say.