There are three things I would check:
1)Is the 'Avast! Enterprise Mirror HTTP Server' service running?
2)Yes, I would setup a Client-Side Updating Task and set a schedule for it.
3)In the AEA, right click on the 'Computer Catalog' (Assuming you use a single rule set for all clients) and select properties.
Navigate to the Policies: avast! Antivirus tree and select 'Updates'
Ensure the following settings are correct:
Update From: EAS
VPS Update: Auto
Program Update: <up to you>
[X] If mirror is unreachable, update from the internet (I don't recommend disabling this to fix your issue, as you want valid definitions regardless of how they are obtained)
You could also check the following, but I'm pretty sure these settings are meant for secondary mirror servers...:
4)In the Computer Catalog Properties, check that the 'Policies: Mirror" settings are correct.
-Mirror From: EAS
-Auto mirror every: 30 Minutes
If you are still getting updates from the internet, then I think you need to start troubleshooting communication between the clients and the EAS.