IIRC Plexamp depends on some level of internet connectivity to get things going. Media should still stream locally if you have your car network set up properly. The only exception might be playback of media you downloaded to the Plexamp client ahead of time.
As for the server, there’s certain means to optimize your setup to reduce / minimize the impact of losing internet connectivity. Though at some point the server and apps need to authenticate your account (e.g. incl. Plex Pass status which is required to use Plexamp). From what I’ve seen, there’s no way to run a Plex setup that’s entirely offline all the time without running into issues sooner or later.
