In San Rafael, the Building Official issues permits for work that builds, alters or repairs a building. The city's rules don't spell out whether a ground cover on its own needs one, so I find out before we start. If yours does, we pull the permit and set up the inspection, which the city lets you apply for and schedule online.
The vent choice matters here too. California's energy rules say an unvented crawl space must have a vapor retarder, a moisture-blocking layer, over the dirt floor, because it keeps moisture from getting in and protects the insulation from condensation. So in a vented crawl space the plastic is a smart move, and in a sealed one it's required.
Under older homes around here, we often find plastic that's torn, shifted or buried under stored boxes, and sometimes none at all. We do the same work in nearby cities like San Anselmo and Novato.