That's tricky question - it depends upon the version of Flash, what network services, what permissions, and some other details, that it requires.  For instance a Flash 9 movie using relative paths to a server not allowing crossdomain fetches would *not* work on a chumby.

Assuming all that is OK, however, you can often view the source of the webpage and search for "SWF" or "swf" - that will lead you to where the Flash movie can be found, although that can still sometimes be tricky.  If the movie will work standalone in a web page, it will sometimes work on the chumby.

So now what? Well, these movies are the copyrighted property of the stations, so you can't just upload them to our servers without getting their permission.  However, if you use this trick, you could have your chumby pull the movies from their server, once you know the URL.