California's energy rules say a crawl space with no vents, anywhere in the state, must have a vapor retarder over the dirt floor, to cut down moisture and keep condensation off the insulation. So if you seal yours, the plastic isn't optional. With the vents open it's a practical call, and general guidance says vented crawl spaces tend to do fine where summers are hot and dry, even with cold, damp winters. Weigh that for your own house.
I won't guess about whether your job needs a permit. The City of Chico's building division is the place to ask, and contractors applying there for the first time need a city business license and a California contractor's license number. If your job needs a permit, we handle the application and inspection with the city or county office that covers your address, and we'll tell you which office that is.
We're based in Chico and work in the cities around it, including Durham and Oroville. A lot of our crawl space calls come from older houses on raised foundations, where the plastic that's down there has been torn up over the years by plumbers and cable installers crawling across it.