No one who earns income is exempt from taxes. A person who earns money working always reports that income on a tax return in their own name. This does not stop you from claiming the person as a dependent if they meet the qualifications.
Your child may be exempt from withholding if this is the first year they work or if their income is expected to be less than $6200. If their income is more than $6200 they will owe federal income tax and must file a return in their own name. They may also owe state tax with different filing limits.
At that income level they can use the Turbotax Freedom edition. Or do it by hand on a form 1040-EZ downloaded from the IRS web site. Good learning experience.
