We shoot around my father’s horses regularly. Their first encounter with it may cause a bit of a jump. They’ll be curious, they’ll probably watch it for a bit, but they’ll ease up to it rather quickly. My father is part of a local city’s mounted unit of their police department. It’s a volunteer group made up mostly of retired officers and a few other cowboys. My dad is just one of the other cowboys and a level 2 reserve with the department. They do a lot of work with the horses to desensitize and prepare for anything. It doesn’t take long to get them used to firing from horseback.
Check out trainer Clinton Anderson.
https://downunderhorsemanship.com/discover/clinton-anderson/
He has a lot of good videos on teaching these big dumb animals. I say that with ok due respect to horses, but they’re big dumb animals. Each has their own unique set of issues, but they’re all the same and the operating instructions work across the board. Sometimes you gotta figure out which buttons need to be pressed, but they all have the same set. How you work with and around them makes all the difference. I like the start with a cap gun for a bit, throw in the .22 and then step up to some pistols. I think you’ll be just fine.
Good luck, and I hope this works out for you! We just picked up 20 acres about a year ago that took almost two years of headaches to land. It’s gonna be an awesome spot to raise my boys and I hope to get a few horses lined up some day. Small steps first though, I owe my wife a house before we get buried in bills feeding and caring for horses.