If you check the forums, you will see lots of posts with people experiencing this problem, why that is I as an avast user don't know, but most are able to logon, though that doesn't help you. I don't believe it is a virus as it wouldn't only be blocking support.avast.com but the main domain not just a sub-domain and that would be true of the forums also, so that is why I don't think so.
Do you have cookies blocked at all a long shot I know.
My suggestion of using the support forums was not to disregard your problem but to get the ball rolling.