Agree with Bladepeek. Supplying him with a bow that fits will help when he begins to really want to hit things. Kids bows are inexpensive; I've found bows for neighborhood kids on an auction site and often people will let them go cheap when they know its going to another kid.
Short light bows are also easy to make. My own son received his first bow at 20 months of age. That bow was made in a matter of hours from a canyon maple shoot, It's 36" long and draws just a few lbs at the ~12" he drew then.
The trick to performance is in the arrows; Easy to make from appropriate sized dowels. Fletching doesn't have to be fancy and can be done by hand with quick-tacking contact cement -"Velcro Adhesive" I like best.
My son's first arrows used 3/16" doweling with .22 casings for blunts. For "sharps" to be shot at soft targets I just sharpened the end of the dowel. His first set was lovingly made, with walnut oil finish, grouse tail feather fletching, and fluorescent red nylon wraps at the self-nocks. He promptly shot them all off into the woods! :( The next set were simply capped with fluorescent orange spray paint and left as raw wood.
His next bow was a 44" long flatbow to be drawn 18" and made from a straight grained maple board and it is a serious shooter with its 3/16" arrows. I needed real weighted points for these as he was really shooting by then. I had to make them with a concrete nail head popped down into a short piece of 3/16" brass tubing. With this set up he could shoot flat out to 20yards, shooting side by side with me. And those little arrows really stick into that foam!
His current bow, at age 10, is a 54" hickory flat bow coupled with 1/4" arrows for up to 24" draw.
His next bow will be an adult bow, with 5/16" arrows. He's still got another year or so with that 54"er though.
Essentially, he's had bows and arrows that fit him right along, and that operate at peak performance for his size. I've made bows as small as 8" long that will sink a comparably tiny arrow deep enough into foam to scare you. I'd never leave one in the woods bc I'm afraid of what the squirrels might do if they ever got a hold of one.