I would say the best person to talk to would be their pediatrician. He or she can make the best determination of their ability, and can guide you on what's appropriate for them.
In general, kids are natural-born runners; they love to move, and as long as you're carefully monitoring their progress and NOT pushing them to do more than they're capable of, and including lots of rest time, it may be possibile. 5k may be a bit much for all running at 6 or 7, but there's no reason your child can't train with you. Kids are also VERY good at telling you exactly how far they can run. My youngest (5) sometimes runs with us when we walk (she has short legs, and a lot of energy.) After about a mile of intermittent running, it's "mommy, I'm tired!" and I end up having to carry her the last mile or so. ;)
A couple of articles that may interest you: www.health.com/health/article/0,,20411921,