My suggestion is to add a separate switch for the fog lights. If you use a relay, then I suggest getting power for the switch from the taillight connection at the headlight switch. That way, you can't forget to have the taillights on when running with fogs, but can run fogs without main headlights.
Almost any modern relay will work fine, IMO. I bought a dozen cheap relays on eBay some 20 years ago and they are all still working fine.
In this situation, I would say a relay is optional. It does let you wire as I've described above, and help extend the life of the original headlight switch.
But a simple automotive rocker or toggle switch can handle the current just fine, and will probably outlast the car. Your call.
After a bad experience some years ago, I'm allergic to a single fuse that can put out the headlights if it fails. I was blowing down a mountain when the lights suddenly went out (for no apparent reason, there was no overload). Could have been a lot worse, but I did have my preteen daughter in the car. So I now have separate self-resetting circuit breakers for each headlight. A single fusible link would probably be just as good.