I used to shoot competitively in IDPA and I've always wanted to try doing it on horseback - but not on Bobo! I came close to buying a gelding that was used in mounted shooting a few months ago and still wish I did. One day I'll give it a try.
The ammo used is made to shoot a distance of only ~20' and at competition, you have to use the ammo supplied by the organization - not your own. Doing it that way makes it safe for everyone. It is not a single projectile such as a bullet but rather something similar to bird shot.
It's done around here. My vet does it and tells me I need to get started. Wanted to for awhile, but slow to start anything that cost money and requires planning ahead.
Training a horse to be calm around noise is just like any other new thing. You can start by firing a .22 when you feed. Call them and fire the .22 and soon they will come to the sound. Be VERY, VERY careful not to do this with spooky horses in a small pen. You won't like what happens next. I could fire a .30 cal from my horse when I lived in colorado and rode in the national forest. Started her with a .22 and her on a long lead rope. If you have a horse with any sense at all, this will be an easy thing to do. Scatter brains take longer.
Also, if you live in town, you might want to go out to the country to do your training. It will keep you from having to deal with those unpleasant people that will show up unexpectedly and uncalled.
I have shot off my horse; usually snakes, armadillos, and prarie dogs. It used to really bother Dobe but he got used to it and now stands there like a champ. Don't know that I could hit anything at a lope though. It would be cool to try. :)