In winter, on the bottom of a humid river valley, I regularly commute in fog with visibility under 100 feet. Not really a problem, just slow down a bit and make sure you're well lit.
Many headlights will throw a wall of glare in front of you that makes them worse than nothing -- you'll want a light mounted below bar height and forward of your face, preferably with a strong top cutoff on the beam to minimize glare. Mine is fork crown mounted, about 6" ahead of the crown.
Tail lights need to be bright in the fog. I have a pair of 1W PDW Radbots that cut through fog pretty well, plus a lot of rear-facing diamond-grade reflector. If you have to ride in the street, stay visible, fog increases the risk of not being seen at intersections and by overtaking motorists if you're not where they're looking for conflicts.
If you have ninja cyclists in your area, be extra wary in the fog, unlit riders are even less visible than usual, and the effectiveness of reflectors is reduced by the scattering of your headlight beam.