What sort of competition? Are you looking at doing trail classes (which are held in rings during "normal" shows or doing things out on an actual trail?
I don't know much about trail classes in shows, but there are lots of distance riding organizations that hold rides all over the country. AERC
is the governing body for endurance, which has organized rides of 25-100 miles. NATRC
is the nationwide CTR (competitive trail ride) organization, which has organized rides of 15-100 miles. ACTHA
does shorter trail challenges (generally 8-10 miles) which incorporate judged obstacles.