Several hours in, this particular bug seems to have been fixed. However, the extension does now seem to sign out very frequently -- e.g., after each time waking from sleep.
Microsoft Authenticator Autofill Chrome extension 2.0.2 stuck on "loading" - Linux/ChromeOS
The latest (2.0.2) version of the extension has not worked on my Chromebook for about 3 weeks. The extension won't load; clicking the extension button just brings up "Loading...". Uninstalling and reinstalling the extension, then re-syncing the passwords, allows it to work just once.
This has been noted by others in many recent Chrome web store reviews (see "Support"), noting the same behavior across several Linux distributions: https://chrome.google.com/webstore/detail/microsoft-autofill/fiedbfgcleddlbcmgdigjgdfcggjcion/related
I've reinstalled the extension in order to report this as a bug in the app.