As an FYI to anyone else who may have encountered this, the recent update to Edge (101.0.1210.32) appears to have impacted Connector.
After the update installed, none of our users could see Connector icons in Microsoft Edge and shortcut key weren't working either.
Something about the update seems to be preventing Connector from running, so we had to roll back to the previous release.
Laserfiche is aware of the problem, but as a short-term workaround we used the following guide to do an organization-wide rollback.
Going back to build 100.0.1185.57 was enough to get Connector working in the short term so we could buy some time.
UPDATE: According to LF support, deleting URL restrictions will also allow the icons to display, albeit with the side effect of exposing profiles regardless of the URL.