The first answer to that question comes from you. Only you know your child and know when he/she is ready to accept the responsibilities and discipline that archery requires. The second part is … it depends on the child. Generally by 8 years old the child has both the small muscle skill and the discriminatory muscle skill that is necessary to operate a bow safely.
Archery requires the use of the small muscles, in the string hand, primarily the middle three fingers, to harmoniously work with the bow hand. The next part is the telling part... can the child release the string hand WITHOUT releasing the bow hand? Developmentally, those two things must be in place. You can either have a qualified archery coach administer this test or you can. Only NEVER use a bow. Use a piece of string, paracord etc approximately the length of your child's wing span, tie the ends together. Have the child imitate the push/pull of the bow hand and string hand. Then request the child release the end of the string that is on his/her face. If your child can release by opening the string hand, while NOT releasing the bow – he/she may be ready to take a lesson. Small muscle coordinating with discriminatory muscle skill.