Hello! I know this post is older but I wanted to chime in! I used to teach vaulting at a summer camp, and actually had to fill the role of the vaulting coordinator for a few months while we were looking for a new one (I was offered the job but I enjoy teaching riding more). Firstly, you will need at least 2, preferably 3 people. 1 to lunge the horse, one to spot the vaulters, and, if you have a 3rd, one to be supervising kids not on the horse and helping them on the barrel.
The vaulting barrel is definitely a must! You teach everything on the barrel first before subjecting the horse to it. Haha.
You will build up some crazy muscles helping kids mount (part of the reason I wasn't fond of teaching it, I love vaulting though.)
I can go into detail on how to teach the mount with the '1, 2, punch, kick' and some beginner moves if you like. We taught the 'riding seat' the 'free kneel' the 'half flag' and simple dismount to the brand new beginners. Eventually they would learn all of their compulsories and then move on to making up their own routines.
We had a Belgian/Quarter cross who was 15hh but WIDE. She was perfect for teaching newbies and she would set a perfect circle, placing her feet in the exact same spot every single lap around the circle. We used to joke that we could tie the lunge line to a post in the center and she would just keep walking that circle all day. It was pretty amazing. Then we had a big Percheron who was about 17hh that the more advanced students would vault on. He required a more skilled lunger. Anyway, good luck with your vaulting program! I'm curious to see how it goes!