In Firefox Add-ons tab, does Avast Webrep show up? If it is disabled, it will be listed at the bottom of the tab. You should be able to enable it there, as you state that AvastUI shows it installed for Firefox.
If it shows enabled in Firefox Add-ons tab, then you may have to right-click on the Firefox Navigation Toolbar and select Customize. Then you can drag-n-drop the WebRep icon to the location of your choice.