In order to get gnome shell search provider registered and active
you need to install the firefox-search-provider.ini and firefox.desktop files system wide.

The locations may be distro specific, for instance Fedora and Ubuntu expect
the files at:

/usr/share/gnome-shell/search-providers/firefox-search-provider.ini
/usr/share/applications/firefox.desktop

firefox.desktop is a system-wide Firefox launcher. It may come with your
distribution or you can use this one. Update name of firefox desktop file at firefox-search-provider.ini
according your actual file at /usr/share/applications.

firefox-search-provider.ini registers Firefox as a search provider.
When the file is correctly installed you can see Firefox as a searchable application
at Settings -> Search at Gnome controll center.

Gnome shell search provider is active only when Firefox is running. When it's active
you can see it as org.mozilla.Firefox.SearchProvider D-Bus service.