Work with your child to help him format his story in a typewritten manuscript suitable to submit to a publisher. If your child isn't good with a word processor, this is an opportunity to teach him how to use a word processor in his writing endeavors.
Consult a market guide that features magazine markets that accepts work from children only. There are many publications that fit this criteria. Help your child choose a market that accepts the type of story she's written. Read the guidelines for submission and go over those guidelines with your child, explaining to her that the guidelines give instructions for the type of story she can submit, how to submit her story, and where to send it. It's never too early to get a potential writer acquainted with the publishing world.
Go over your child's story one more time for mistakes. Just because the publishers you're submitting to publish stories by children, it doesn't mean the stories can have be sloppy and full of errors. Practice the same careful editing you would for a manuscript from an adult. Involve your child in the process.
Wait for a response from the publisher. Most guidelines will give you a response time. If one isn't mentioned, wait at least two months before writing a follow-up letter to check on the status of your child's story. If the story doesn't get accepted, encourage your child to try another publisher. Explain that the rejection doesn't mean the publisher didn't like his story, just that other children submitted stories as well and that the publisher doesn't have room to publish everything.