That would indeed be your subwoofer. There are a few things that could be wrong.
-The subwoofer is partially blown (damaged speakers can make all sorts of noises)
-The amplifier has something wrong with it
-There is a wiring problem between the radio and the sub amp.
-There is a problem with the radio head unit itself.
Your subwoofer assembly is installed behind the rear right quarter panel trim. It's a bit of a pain to remove it, but you'll need to take that whole trim panel off to troubleshoot the subwoofer. You'll find the black plastic subwoofer enclosure with the amplifier screwed onto the side of it.
First, I'd try hooking up a different speaker (easiest and cheapest thing to test). If the problem is still present, you can follow the audio wires from the amplifier to the radio to check for damage. There are five wires (power LG/VT, ground BK/LG, audio signal + BN/OR, audio signal - RD/BK, and audio mute DG/VT). The 3 audio wires should run in a shielded bundle in the wire channel on the passenger side. Check for proper power on the power lines.
If all that checks out, you'll need to try swapping in a new head unit or amplifier. The amplifier was shared with several other ford models, so you should be able to find one at a junk yard - just make sure it's the same model number. The head unit might be a bit harder to find, since only that version has a subwoofer output.
You "may" be able to hook up an external audio source to the amp to test it - I've never done it, but there's no reason I know of that it wouldn't work. I don't know the exact role of the "audio mute" signal (i.e. whether it needs to be connected to +12V or ground on it to be "unmuted").
You cannot attach the subwoofer output from the head unit to an aftermarket amplifier. There is noise in the signal that the stock amplifier filters out.
