No. Most people apply for a child's card as soon as they get the birth certificate. The child is not required to have one but it is required to claim your child as a dependent on your income tax return. Also should you wish to open a bank account for the child; buy savings bonds for the child; obtain medical coverage for the child; apply for government services for the child; obtain a passport; or enter the child in school (some states) a SS card is required.
Thank you for the info.