I had the same problem with ZA 5.5 (home). I think what happened was that its request to permit access by ashWebSv popped up at the same time as several other warnings/notices and got overlooked. So I could access mail, IM’s, etc. just fine but not anything via my browser.
Fortunately, going into ZA’s control panel and altering avast’s “?” status manually cleared that up nicely.