Can I hire non-licensed architecture firm for residential project?
No, you do not need a licensed and/or registered architect for a residential project in California. 

You can check with the State licensing agency. Here: https://www.cab.ca.gov/

The building department can require anything, including peanut butter on the roof. (If they do, there’s an appeals process, which they’d loose on the peanut butter issue.)

However, if they require seismic calculations, footing calculations, plumbing calculations, electrical calculations, setback requirements, etc. your designer can hire a local engineer and/or architect to prepare such calculations. 

Make certain you know what your design firm is responsible for...approved plans and specs suitable for a building permit? AND what is the fee for such plans and specs and when they’ll be completed and suitable for submission to the Building Department. 

Also, are they going to assist during construction? If not, what is their fee (hourly) to answer the contractor’s questions. 

Are they helping pick the colors, size of HVAC unit(s), plumbing fixtures, vents, etc.