How do I find out who the insurance carrier is for errors and ommission insurance for a lawyer in Caifornia

You cannot sue the attorney's insurer directly unless you have a judgment. Check your fee agreement to see if it discloses whether the attorney has malpractice insurance. The attorney does not have to disclose the insurer or policy limits except through discovery after a lawsuit is filed.

If the attorney is covered and if law suit is filed, very likely it will be tendered to the insurance carrier. Likely through discovery phase you could determine e coverages etc. As mentioned by attorney Kane, there is no obligation to disclose the identity of the carrier but only if there is insurance covering the attorney which would be reflected in the retainer agreement.

I have no idea why you would need this info, but you may want to contact the State Bar.