GNOME shell extension doesn't work with snap version of web-browser

You don't need to install extensions from extensions.gnome.org as they are also available in 'Ubuntu Software' app.

Launch Ubuntu Software, click add-ons, click extensions tab. Here you are able to install extensions.

Since you're using the snap version of Firefox 64.0.2, it's not unexpected given various restrictions of snappy.

This issue is already raised in snapcraft forum: https://forum.snapcraft.io/t/firefox-snap-is-unable-to-access-chrome-gnome-shell-connector/4562, but unfortunately no solutions were provided there.

Similar issue can be seen with the Chromium snap too: https://forum.snapcraft.io/t/chrome-gnome-shell-does-not-work-with-chromium-snap/3377/2
(see the bug-report)