InvisionFree is a hosted solution. I assume this means you do not have access to the server files. This means you can not customize the system (beyond any admin tools they give you).
By the way, the ".php" at the end of many InvisionFree URL's indicate it is a PHP based system.
You can't create a shoutbox with javascript assuming you want the shouts to persist between sessions and be shared between visitors. That is, if I enter a shout, you want to see my shout when you access the site.