Are you talking about the corporations legal corporate name - you search for that on the California Secretary of State's website - see link below.
Or are you talking about a fictitious name (dba) that the corporation is using OTHER than its legal name. Because a corporate name is not a fictitious name, it's the corporation's real name.
Fictitious names are registered county-wide and other than a hit and miss search on good old' Google, there isn't really a state-wide public database of fictitious business names.
You don't tug on Superman's cape; you don't spit into the wind; you don't pull the mask off the old Lone Ranger; And you don't get legal advice from a free Q&A page on the Internet. The above is a general statement of the law or just my opinion. I am not saying whether it applies to your situation or not because I don't know the details and youâ€™ve not hired me as your attorney.
You should determine the specific county in which the fictitious business name was recorded and search the name in that county's recorder's office. Most if not all counties in California have website where you can search the fictitious business name or names.
This answer does not constitute legal advice and is provided for informational purposes only.