If a person is a minor (under 18) can their parent...

To replace a Social Security card for a child:

Step 1: Gather documents proving your:

• Child's Identity
• U.S. citizenship if you have not established the child's U.S. citizenship with the SSA
• Immigration status if the child is not a U.S. citizen.

Step 2: Gather documents proving your:

• Identity
• Custody/relationship for the child

Step 3: Complete an Application for a Social Security card.

Step 4: Take or mail the completed application and documents to your local Social Security office or your local Social Security Card Center.

All documents must be either originals or certified copies by the issuing agency. SSA cannot accept photocopies or notarized copies of documents. Any documents you mail to SSA will be returned to you along with a receipt.

http://ssa-custhelp.ssa.gov/app/answers/detail/a_id/291