I think that Browning has a Micro-Hunter Rifle to fit Younger/Smaller Hunters, and as several mentioned here its best not to rush a Young Hunter and if they grow into a Rifle and learn to shoot it well and accurately it will be better for them as the get older.

If You hand-load Your own ammo You could get a .243 and possibly lighten it up so the recoil wouldn't be as bad....or simply start them out in a .22LR to learn good shooting habits then move up to the .223 Rifle if allowed to use for Deer Hunting in Your State?Its better to be accurate with a smaller Caliber Rifle than to miss with a larger Caliber.