is there a state or federal law that requires a U.S. citizen to have a social security number, at what age is a SSN required

There is no law that requires a person to have a social security number. However, without a social security number, a person is not eligible for many government programs including:

Internal Revenue Service for tax returns and federal loans;
Employers for wage and tax reporting purposes;
States for the school lunch program;
Banks for monetary transactions;
Veterans Administration as a hospital admission number;
Department of Labor for workers’ compensation;
Department of Education for Student Loans;
States to administer any tax, general public assistance, motor vehicle or drivers license law within its jurisdiction;
States for child support enforcement;
States for commercial drivers’ licenses;
States for Food Stamps;
States for Medicaid;
States for Unemployment Compensation;
States for Temporary Assistance to Needy Families; or
U.S. Treasury for U.S. Savings Bonds


The practical effect is that unless a person has no need for banks, insurance, school, state licenses or work, the person will need a social security number.

There likely are still some US citizens who do not have social security numbers.