Ok, it seems a few people have different ways of doing this. I have a way that uses NO parts, requires NO bending, NO cutting, NO splicing, and a bare minimum of time and tools. Are we ready?
This works on my 2012 4x4 Double Cab TRD Sport. It may not work on older trucks due to changes in the wiring and fuse configuration.
First, remove the fog light relay
Next take the relay box off and flip it over. There is a cover on the back side that can be taken off to access the wiring.
Finally, locate the two wires indicated by the RED ARROWS. This picture shows the wires already switched.
The connectors can be pushed out from the TOP (relay side of the block) with a jeweler's or precision screwdriver. There are little plastic tabs that hold these in. It takes a little practice and patience to push them out but it isn't that difficult.
Once you get both wires loose, simply swap them, reinstall the bottom cover, relay and top cover and you're ready to go. Your fog lights with come on with the switch anytime your ignition is on.
This works because the fog light switch switches the negative side of the circuit to the fog light relay coil. The wire that was swapped has +12v on it anytime the ignition is on. This is used to power the positive side of the relay coil. This is too easy to do!