Protecting Horses from Soring Act of 2021 This bill revises requirements concerning the inspection of horse shows, exhibitions, sales, and actions for sored horses (e.g., horses suffering from pain when moving due to substances or devices placed on their limbs). Specifically, this bill replaces a requirement for the Department of Agriculture to prescribe regulations governing the appointment of inspectors with a requirement for the Horse Industry Organization, which is established by this bill, to appoint inspectors.