Preface
If you are running wayland and using a window manager, you need to set the QT_QPA_PLATFORM to xcb in your environment variables. Otherwise Teamspeak3 will not run. You may override it temporary by running Teamspeak3 or any other qt application with the --env=QT_QPA_PLATFORM=xcb.
Example
flatpak run --env=QT_QPA_PLATFORM=xcb com.teamspeak.TeamSpeak3