Local Chris Hoyle allowed his son Nathan to play in a tournament a couple of years ago when he was three. I think three other kids under 9 played with him (all 7 or 8, he smoked them which was impressive) but the TD set up the division as a one-round group, which I think is important for kids that young. If you can get a TD willing to put a responsible adult with them and limit it to one round... do it as soon as he understands courtesy and sportsmanship.
When they ask, "Can I play a tournament?" it is time to see if they can go two rounds, stay on task and have fun. I also made my girl recruit some friends to play so that she would have a foursome or threesome. After we had played some casual rounds, I took the girls out for a "practice tournament" to make sure they were ready.
Keep it fun. If you walk with your kid's group don't be badgering him, or hectoring him, or giving him coaching he isn't asking for. If they are having trouble keeping accurate scores, make tally marks on a scorecard as they throw and keep them honest on their scores.
Kelsey is 18 now and still loves the game. We played a practice round in a steady rain last night just to practice our rain game. Not too many kids would humor their dad that way.