To be strictly accurate, she was a volunteer with the Pony Club, not a member thereof - you have to be a young person to be a member of the Pony Club. Volunteers are largely unregulated beyond a DBS check and some very basic first aid / safeguarding / CPD. Obviously some volunteers / instructors have better qualifications, but it's by no means universal. That's an issue with the Pony Club that goes beyond this story.
Regarding this incident, her behaviour was reprehensible and she certainly isn't the sort of person I would want teaching children in any capacity.