I'm not sure about the VPN, just Avast straight out of the box with some Exclusions.
If you look at Avast UI > Settings > Components > SecureLine VPN...
Is it enabled?
If you're using a VPN and connected to a server (different country), the IP address would be different resulting in the banking site asking you to verify to ensure it's you.
If not, just ignore my previous comment and try this:
If you go to Avast UI > Settings > Components > Web Shield and disable the shield, does the same issue persist?
If that doesn't work, you'll need to disable each shield manually to figure out what component is behind the issue. Alternatively, you can disable Avast completely via the tray icon and see if the issue persists.