The governing body for national eventing in the UK is British Eventing.
Eventing photography is very popular but requires in depth knowledge of dressage, showjumping and cross country photography to do it well. Be aware that most horse trials have resident event photographers.
British Eventing don't typically have anything to do with assigning photographers to events. This is usually done by the organiser of the event, who will also determine the rules which press photographers will have to abide by if they are to receive accreditation.
Most events use the 90-day rule, which prohibits "press" photographers from providing images to competitors in any way for three months after the event.