It doesn't matter where the company is incorporated, you will have to register it as a foreign entity in your state (in this case, CA).

Take the time to read about this before doing anything. One impulsive action could lead to thousands of dollars in annual fees and even more in filing costs (accountants, lawyers).

With C-corp in DE registered, do we have to register with CA if no office/people in US?

Everyone's situation is different. That would be a good question for an attorney or accountant.

See my other comment about having a 'nexus' in California (which is the topic here).