There are many products designed for smaller bodies. Children, like adults, need safety equipment. Even shooting outdoors, hearing protection is very important for young ears. Eye protection is also so important. Many .22 rifles are available in scaled down sizes. Pellet rifles are also good for smaller hands. Scan the internet and you will find small children successfully, safely and accurately shooting full size firearms from a bench rest under the very watchful eyes of a parent.
As they grow, family trips to the range, or hunting, can be a wonderful was to bond. Learning to master the skill of shooting also gives confidence, remember your first bulls eye? Knowing gun safety can protect them if they encounter a firearm away from home.
I do encourage you to work with an instructor in teaching your child. So often, even experienced shooters, forget things they take for granted, such as dominate eye or stance. They know to do it, but don’t remember to explain it.
Teach your kids to handle firearms safely and respectfully and your family can enjoy years of shooting sports together.