I having the same problem as you. New install on three different computers (2 iMac's and Macbook Air) on Mountain Lion. It worked well in Lion on all three, with no problems logging in or out of Mac App Store. I have done and update from Lion, erased hard drive and started fresh install with same results. I connected straight to the modem and through Airport Extreme, with the same results. I am on the phone with Apple and they are unsure what is causing the problem so far. They say it could be internet provider, but that would make no sense if it worked fine when Lion was just installed, and Lion login and out with no problems. The tech person, John (high level tech) is being very helpful. I sent him the error file so they can analyze the problems. By the way, I am using Comcast(Xfinity). I want to see of other people that are having this problem, what network they are on, to at least narrow that possibility out of the equation.
My guess is not a lot of people who installed Mountain Lion know that this is a problem. When first installing Mountain Lion, it ask to put in your Apple ID. So when it starts up, the Mac App store is already logged in. Not many people log off, unless there is multiple users.
So, if anyone else hasn't tried yet, log in and out of Mountain Lion Mac App Store and see if it is crashing also.