The avast update doesn't require your browser FF to be loaded to be able to update.
Well I have been using Outpost Pro for many years and no issues, however, I never have the anti-spyware module as it is just duplicating scans (especially at boot). Nor do I have any other modules other than firewall related as I like my firewall to be just that a firewall.
It may be that OP is not fully loaded and may block outbound connections until it is. You could try delaying the auto update check to enable everything to be loaded.
You need to edit (using notepad) the [InetWD] section of the C:\Program Files\Alwil Software\Avast4\Data\avast4.ini file and add the following line (depending on your connection type enter one or other of the Bold lines below):
Dial-up connections, add this line:
RASWaitSeconds=120Broadband connections, add this line:
AlwaysConnectedWaitSeconds=120[InetWD]
AlwaysConnectedWaitSeconds=120
Or
RASWaitSeconds=120
When complete save the changes, avast's self-defence module will ask for confirmation, etc. answer Yes.
The figure is seconds and the above equates to two minutes, you could try that and adjust upwards if required, 180, 240, etc.
Edit:
Something you could try first - avast Program Settings, Update (Connections), set the method you use for connection, one or the other. Click the Proxy button and set the proxy to Direct connection (no proxy) if you don't have to connect by proxy.