Did you tell RHHT that you would be offering your images for sale to the competitors? If they had no problem with it then your are technically in the clear, but, yes, you are stepping on the competition of an OP who invests a lot of money and time, usually sun up to sun down to be there and provide a service to competitors. It is a free market and if management lets you, sure, you are allowed to come in and shoot a few people, undercut OP pricing and see if you sell images, but there’s an ethical line there. This is one of the reasons many horse show photographers have stopped shooting events.
If you want to learn to shoot the equine stuff and get better at improving your timing and exposure, I think you will find that you’ll have a much warmer reception from the few remaining horse show photographers if you don’t try and sell your event images - posting them to galleries or FB for viewing, but not offering them for sale or free, unless you want to be the show OP - purchase insurance, get the booth set up, get the staff, and do the whole gig, so you’re not undercutting someone else’s livelihood. We’re struggling enough with online proof theft and everyone’s barn mate’s and Mom and Dad’s with DSLR’s that make it hard enough to make a living doing this.
I no longer shoot horse shows - only portrait sessions and weddings.
Here’s a couple threads that make for interesting reading if you have a lot of time on your hands regarding issues related to the horse show photography industry:
http://www.chronofhorse.com/forum/showthread.php?301055-Where-are-all-the-photographers
http://www.chronofhorse.com/forum/showthread.php?371032-Independent-photographers-at-horse-shows-quot-legal-matters-quot
http://www.chronofhorse.com/forum/showthread.php?390583-Private-Horse-Show-Photographer
http://www.chronofhorse.com/forum/showthread.php?406522-Theft-of-Photographs-from-Show-Photographers/page6