I have never needed a birth certificate to register to vote.
You could use a SSN from a person residing in another state.
Or, according to this form: https://www.eac.gov/sites/default/files/eac_assets/1/6/Feder..., "If you do not have a current and valid driver license or non‑operating identification license or a social security number, please write “NONE” on the form. A unique identifying number will be assigned by the Secretary of State."
I have no idea what verification is done on the back end, but registering as a voter in CA is as simply as filling out a postcard and sending it in.
I'm guessing there isn't much verification, as I've heard of immigrants accidentally being registered to vote because they checked "Yes - I'm a US citizen" on a driver's license form.
You could use a SSN from a person residing in another state. Or, according to this form: https://www.eac.gov/sites/default/files/eac_assets/1/6/Feder..., "If you do not have a current and valid driver license or non‑operating identification license or a social security number, please write “NONE” on the form. A unique identifying number will be assigned by the Secretary of State."