For whatever reason, my installation of Ubuntu GNOME Remix 12.10, didn’t have Quicktime support added to Firefox by default. To make a website work near enough as it was intended, this needed to be remedied. The solution was to issue a command to install the missing software:
sudo apt-get install mplayer gnome-mplayer gecko-mediaplayer
Mplayer is the main component here but running the above command adds all the required supporting packages too so that a Firefox restart was all that needed to get things going.