I never had the opportunity to show my horses at the Big E, I didn't do much 4H with the horses, but I did show dairy cows. When I went I was part of a state team, they had a selection process. I want to say the horse show was similar. Whether there was a qualifying show or it was a point accumulation, or an application process I don't know, but I would think your state 4H office will have all the details and should be able to clarify things for you.
The Big E is was so much fun! We had no option but to stay in the dorms but it added to the experience. I wish your daughter luck in qualifying, it will be worth the effort!