As grandfather of 8 who range from ages 2-7, and archery teacher of kids from 8-14 (and sometimes their parents), the youngest I have personally seen be able to shoot the bow with any sense of purpose was 5. That "purpose" didn't last very long, and the youngest we allow in our classes is 8, although we don't check ID's, so it's possible some younger ones might have slipped in.
I have found that kids from 5 up seem to be able to manage a 15 pound bow. 1516 aluminum arrows work well and are light enough to make it to a 20 yard target without too much arc. Of course, the usual shooting distance for a 5 year-old would be a lot closer than that, maybe 10 feet. I recommend a recurve, as you want to get everything possible out of a bow that light. Depending on individual strength, the 15# bow should work well for them up to about the age of 10, so you don't need to buy a piece of junk. I think this is because they naturally draw it longer as they grow, so they might be pulling 10# when they are 5, and the full 15# when they are 10.