Is there a way to get the "Plasma Integration" plugin working for flatpak sandboxed browsers?

Look for native messaging. It needs a flatpak portal or some weird workarounds.

Other addons like kde connect, keepassxc-browser, zotero, umd and others are also affected

The way I have it, is that I copied org.kde.plasma.browser_integration.json from /usr/lib/mozilla/native-messaging-hosts/org.kde.plasma.browser_integration.json (for Kubuntu 22.04, might be elsewhere for you) to ~/.var/app/org.mozilla.firefox/.mozilla/native-messaging-hosts and then changed the path to a shell script that calls the original executable with flatpak-spawn --host. Of course this kind of breaks sandboxing since you are allowing the browser to access programs on your machine but it works.

So I have: org.kde.plasma.browser_integration.json :

{
  "name": "org.kde.plasma.browser_integration",
  "description": "Native connector for KDE Plasma",
  "path": "/home/username/.var/app/org.mozilla.firefox/.mozilla/native-messaging-hosts/org.kde.plasma.browser_integration.sh",
  "type": "stdio",
  "allowed_extensions": ["plasma-browser-integration@kde.org"]
}

and org.kde.plasma.browser_integration.sh

#!/bin/bash
flatpak-spawn --host /usr/bin/plasma-browser-integration-host "$@"

Don't forget to chmod +x the shell script.

I have the same for the KeePassXC extension.