I released an update that fixes compatibility with Firefox 101.0 yesterday, but it hasn't been approved by Mozilla yet. Until it's approved, you will have to manually download the native program from GitHub: https://github.com/filips123/PWAsForFirefox/releases/tag/v1.4.2

Flatpak (and Snap) currently cannot work because of sandboxing limitations (and the same is true for any extension that uses native messaging API). This cannot be fixed until native messaging is supported by Flatpak and Firefox. You will have to use normal Firefox.