Native Connector is supported now
I'm using Ubuntu 22.04.3 and I can confirm that the Snap version of Firefox does work with the Gnome Extensions website.
All I had to do was install the connector like usual:
sudo apt install chrome-gnome-shell
And then install the Firefox add-on when prompted by the extensions.gnome.org website.
After reloading the page there was an OS pop-up message to grant permission to the native connector.
References
I couldn't find a whole lot of information on this, there's far more search results that say "uninstall the Snap version" than there are that say that it now works just fine with Gnome Extensions. I did find this blog post though, which linked through to this call for testing on the Ubuntu Community Discourse site.
Screenshot from UbuntuHandbook:
