Okay... here we go!
So it looks like an LED switch (which may be burnt out). The 3 prong toggle switch is a single pole-single throw with LED. The ground circuit is
only for the LED.
One of the wires goes to Ignition, that powers the LED and transfers power to the circuit when the button is pushed. (picture may be different as far as the toggle switch is wired depending on what prong does what - should be labeled, GND, IGN, LOAD - or you can take a DMM to it)
The last prong as you may have guessed goes to the relay, this is a separate usually lower amp circuit that switches/clicks the higher amp circuit in the relay on, this is the power side of the switch. The ground at the relay is
only for the toggle switch circuit. (Like in the pic below)
The inline fused wire at the relay is a separate power source/circuit for the fog lights them selves.
A relay uses a low amp circuit to switch on/off a High amp circuit. You have a relay that is the same concept as the Bosch one I posted earlier.
If there is no fuse in the Fog light circuit then the lights wont turn on BUT the relay will still switch because of the separate power and ground that controls it.
If you put a fuse in the holder and connect that wire to a power source the fogs should turn on if they are correctly grounded. like in the pic below (make sure you check that all the wires are correctly hooked up before you do this. There might have been an issue with them before - which may be why its disconnected).
Follow this picture and make sure everything is wired correctly, then you can still use your fogs, and just put in another switch for the subs somewhere like under the dash, or next to that one
Be careful and good luck, this is not exact as I can't look at everything in person so make sure you know where all the wires go. oh and don't trust aftermarket wire colors. The previous owner may have used the same color wire for different things, I've seen some people use the same color wire EVERYWHERE for their "aftermarket" jobs (makes it really confusing).... having said that I hope I wasn't too confusing.
I can clear up anything you need me too.