I can see how this might get murky, since sometimes a small barn might not officially be a business (even an LLC) but is still renting stalls/has business related expenses. But unless the barn is paying the farrier directly for all the horses serviced, it’s still an individual transaction (like a housekeeper or plumber) between you the horse owner and the individual farrier.
In terms of tracking payment, it’s up to the individual to report it. Obviously, a farrier might underreport income when he can get away with it, although with more people using electronic payments, there is often more of a paper trail. Also, if a business is in business for years yet mysteriously has a tiny income despite having business expenses, I assume that would be a red flag.
Being audited sounds like such a nightmare, reporting every penny seems like the more preferable option!