Breed Show No, that would mean that they would have to be NSH, ASB, Arab or AQHA etc etc etc forever.
Open shows any horse can compete, Rider's usually need to belong to the association whether that be national or Local. (i.e. USEF, USHJA, PCHA, GSDHJA, SAHJA, CSHA, etc etc etc).
Find a show you want to do and call management to find out what you need to do in order to compete at it (like joining associations).
Gordon Wright once told a student to take up swimming because he would never be a good rider, that rider was George Morris.