Ubuntu 18.04.6, Firefox 106.0.2, flatpak 1.14.0 from the official flatpak PPA
gnome-software 3.28.1, gnome-software-plugin-flatpak 3.28.1-0ubuntu4.18.04.15
flathub repo set up and working
I have verified all of the above as working by installing flathub packages, both from cli and gnome-software.
The firefox extension seemed to install without issues, but when I click on an install button on a flathub page - nothing happens. No process starts. Nothing at all. Even if I pre-launch gnome-software, nothing happens.
I don't know much at all about firefox's plugin architecture, javascript or json, but if you tell me what you need I'll be happy to try to help troubleshoot this.