I vaulted competitively all through my teen years (in Germany) and taught a group of little girls after that.
First of all, it depends what your goals are. At our barn, vaulting was often the first contact with horses for many of the girls, and a lot of them moved on to riding when they were old enough. We took vaulters from about 5 or 6 and riders starting at 10, so the typical kids were 5-10 years olds. It was also a good way of recruiting talented kids for the competitive groups at our club.
Try to not make the groups too big, but I would not do private lessons. For one, it’s really, really exhausing - for both the vaulter and the horse. And to me it’s actually one of the big advantages of vaulting over riding that it is a collaborative team sport. If you have a bunch of single vaulters, try to get three or four in one lesson. That being said, be aware of the dynamics in your group. Don’t allow drama and ALWAYS be fair. Pre-teen girls are incredibly sensitive to that and it is very important that there is trust within the group, especially when you are starting figures that include lifting.
At our club, ALL the kids had to be there half an hour before the lesson to brush the horse and tack up together. For one it is important for anyone who wants to deal with horses to know how to halter, brush, put on the surcingle, boots/polos and bridle. Also, if you don’t make everyone do it, the same handful of kids will do the chores every time and it will breed resentment in the group.
It is really helpful to have a second adult / instructor, so one can concentrate on the horse and one on the kids. The instructor that is not lunging can also spot while the kids try out new exercises.
Try to pair up a new kid with one a bit more experienced. That way the new kid gets integrated in the group right away and you have an extra set of eyes on the newbie. It also gives the older kids a bit of responsibility.
I always took about 10min for warm up and tried to make it fun. Generally, the kids were there to be on the horse, so that usually wasn’t their favorite part. Do some running games, e.g. everyone is in a row jogging slowly, last kid runs up and takes the lead. Warm the horse up at a trot, let the kids run up to the horse, give it a good pat, count out loud the number steps the horse is taking, and then run out of the circle. Stuff like that. There are a lot of books with good ideas for exercising with little kids.
Then a bit of stretching and some trust-building exercises, which works well in pairs.
Just like with training a young horse, it’s best to build from easy to harder stuff gradually and finish off with a success. I usually started out with the compulsory exercises (or parts of them if the kids are not quite there yet), then build to something that we’re working on, usually some freestyle exercise, or scissors/flank, since they seemed to require the most amount of work, and finish off on something fun (hanging exercises, rolls, shoulder-/handstands, stuff like that). Even if you are not planning on competing, it helps to have a general goal in mind to work towards, e.g. a demo at the barn christmas party or something.
I always liked doing two- and three people exercises on the barrel and only transferring them to the horse when they work pretty well on the barrel. They build team spirit and trust and look flashy, but they are also quite hard on the horse. You will need an adult to help out at the barrel though.
Also, be really careful with your horse. Lameness issues are incredibly common in vaulting horses and a good horse is so hard to come by. We usually lunged half of the lesson to the right and used Vienna reins instead of side reins when possible. We also made sure the horse got used in 2-3 vaulting lessons a week max, and got ridden well in the meantime (dressage, cavaletti, hacking out).
I’m pretty excited that vaulting seems to be taking of more in North America. It’s a great sport and deserves more attention!