No, it’s not.
IE is only needed to display help files (like a lot of other applications have to use to).
avast can be used with any browser and will update with them. Firefox works perfectly.

Make sure your browser is not set to “Work Offline” (this option is generally in the File menu). If it doesn’t help, try switching the proxy settings from “Auto-detect” to “No proxy” (I’m assuming you’re not connecting to the Internet via proxy).

Left click the avast icon > Settings
Update (Basic) > Details > Proxy