Dear Ms. Collins,
Thank you for posting your request on History Hub!
The Social Security Administration (SSA) does not create or maintain birth certificates. Birth certificates are not Federal government records but are Vital Records created and maintained by State and/or local governments.
We suggest that you check Family Search for information about how to access California Vital Records and also Ancestry for Birth, Baptism & Christening Records. There may be a fee for using Ancestry. However, you may view these records online at one NARA's facilities for free via a NARA PC. For the nearest NARA location, please consult our web page at https://www.archives.gov/locations/. Some of the images from Ancestry sometimes are available on Family Search for free.
We hope this is helpful. Best of luck with your family research!