Casey Weederbot - 2013-01-20

One possible solution would be to tune a media player to that station and then broadcast that through IDJC.
You'd do this using qjackctl's handy patchbay, wiring the player's audio through IDJC.
Actually, looking now, IDJC >= 0.8.8 can switch between jack sources natively, which is very nice.
You would be doubling the delay, as your broadcasting rig would have to wait to receive the primary stream, and then your listeners would have to wait that long again, but depending on the stream this might only be a couple of seconds.

If you're running jack on alsa (probably, this is default), you should be able to make use of alsa plugins to create virtual inputs out of outputs which you'd ideally be able to select under IDJC's new Jack Ports dropdown, but I'll need to experiment it some more, as at the moment my asound.rc settings aren't being picked up at all.

Some reference material:
(for configuring alsa plugins) http://alsa.opensrc.org/.asoundrc
(for easy graphical configuring of jack (qjackctl is also in most repositories)) http://qjackctl.sourceforge.net/