Has her horse shown before, and recently? If so, what if you take her to a little local schooling show that has appropriate classes she could enter (only like 3 or so, and kept *well* within her zone of competence). She will not need fancy clothes for that sort of leetle introductory 'fun' show - you can call the show management and find out what, in practice, she should wear. Look for showbills on feedstore and tack shop bulletin boards and places like that.
If not, I would suggest lessons (at a lesson barn, on lesson horses) and work up to going to a similar fun schooling-type show on one of the lesson horses. (You DO NOT want a kid's first show experience to be on a horse where there are significant question marks how it will react to a strange place, show atmosphere, jillion horses, etcetera. Really truly big-time).
This will give her a taste of generic horse showing, and you can see what she thinks about it. Particularly the part where you do
not win ribbons most of the time
Be aware that if she wants to get 'into' competing, you are going to need a truck and trailer, a nontrivial amount of money (for coaching/lessons as well as for clothes, tack, entry fees, gas, etc) AND you will lose a significant number of weekend days on which you would otherwise be able to work or relax.
FWIW, I would try to stay
away from barrel-racing for now, or any other fast rough-and-ready results-oriented discipline. Reason being, in those disciplines it is REALLY REALLY easy to learn bad habits if you don't have good habits already firmly installed (even if you do, in fact!) and it is EVER so much harder to replace bad initial habits than it is to install 'em correctly in the first place. BIG TIME.
The ideal thing, to my way of thinking, is for a kid to have a good long period of lessons (preferably on a variety of horses - you can't really learn to ride very well on just one) taught by someone who promotes good horsemanship, a good following seat, good body control, and relaxed happy stretchy horses. This includes some (but by no means all!) instructors in hunt seat, basic dressage, and basic western riding. If she learns a good, effective, strong-yet-relaxed-and-sympathetic seat on a variety of horses, when she is in her teens she will be able to make the transition gracefully to whatever specific discipline catches her fancy.
JMHO,
Pat