Often the first real exposure a young poet has to a larger audience is within his school community. Annual contests that celebrate holidays and national observances are often sponsored by schools. Contact the school your child attends and ask for a calendar of events with an eye to the would-be poet. Rewards can include gift certificates, schools supplies, bonds, cash and scholarships.
Check out poetry contests run by your community and state. The National Federation of State Poetry Societies, Inc. offers insight to the world of poetry, including links to state poetry societies, upcoming events in the world of poetry and state-sponsored contests (see References).
Look at the clubs with which your child already has an affiliation. Organizations like the Girl Scouts and Boy Scouts of America have magazines that sponsor various contests throughout the year.
Literary magazines for children are a good place to look for kid contests. Some magazines sponsor their own contests, but they also list contests as part of their news forum. A great tool for any writer is the Writer's Market (see References). In addition to a specific section on contests and awards, it also lists publications that accept and pay for poems written by children.
Although you need to pay particular care when allowing your child to enter any contest online, there are reputable organizations that offer money for poetry in contests (see References). Beware of exorbitant entrance fees and do your own research before allowing your child to submit.