Newborn Automatic Number Assignment
The hospital where your child was born may potentially offer you the ability to apply for your child's social security number via an application called Newborn Automatic Number Assignment (NANA).
In surrogacy cases, however, using the NANA will indefinitely lead to linking your child's social security number to your surrogate. Once this is done, it is very difficult for it to be undone.
Social security application
Once you receive your child's birth certificate, you may apply for the social security card by mail or in person at your regional social security office. The application for a newborn social security number can be found at http://www.ssa.gov/online/ss-5.pdf.