Most incorporated cities in Fresno County, including the City of Fresno, operate their own building and planning departments and issue their own permits. The County Planning and Development Department handles unincorporated areas. Various overlays such as flood zones and fire hazard severity zones may require additional review.
How this fits with the state
The state authority here is California Contractors State License Board. The county file does not replace that — it adds the permit office, local overlays, and the actual filing path.