A relay is not absolutely required, but highly recommended. The fan draws a considerable amount of current, and without a relay, all that current has to flow through whatever controls you use. For example, if you want the fan to automatically shut off when you turn the key off, it would be drawing current through the ignition switch.
I don't know if the factory horn relay would be a problem or not; but I know the aftermarket horn relay that I used for headlights many years ago turned out to not be continuous duty and eventually the coil burned out.
Besides, you probably don't want the horns sounding whenever the fan is on.
Just add a proper fan relay and be done with it. DeRale sells a nice kit with a relay, a temperature probe that just fits into the fins of the radiator, and an adjustment for kick-in temperature. It also has an external input (intended for AC) that you can use to have a (low current) override switch on the dash, if you like.
https://www.amazon.com/Derale-16759-Adjustable-Fan-Controller/dp/B000CN4XZI