Most writing contests have a fee to enter because they need the fees to pay for the prizes. In fact, a lot of small magazines need fees from their writing contests just to stay in business.
A few well funded magazines or publishers may put on contests to find new writers, but this is rare because they already receive far more manuscripts than they have time to read. These contests are usually not open to children.
The places you are most likely to find contest with no fee that are open to children is your local library or town council or other community organizations or charities. Children's magazines may also occasionally run contests.