Start at the socket and use a volt meter to check for power/voltage. If no voltage at socket... probe wires just a couple inches behind the socket. If you get power the socket is most likely the issue. Doesn't necessairly mean socket is bad,,, just sometimes the o-ring seal goes bad,,, and socket gets corroded.
Spray socket with some battery terminal cleaner and let it sit for a few minutes. Then blow out with compressed air,,, and then put some dialectric grease in the bulb socket. Plug in... and see if it works. If so... good to go. If not,,, you are probaly going to have to to to your local parts store and find a replacement "pigtail" harness plug for the LS foglight with 2"-3" wires coming off it.
Solder and heat shrink to factory harness, (or crimp and electrical tape)... and you should/might be good to go. Depends on if you found power just behind original plug,,, and if battery terminal cleaner didn't fix the problem on the original socket.