Plex had to remove posters for some movies due to DMCA claims by the rights holders. This can be checked by finding the movie at watch.plex.tv, as there will not be a poster there either. I’ve not seen posts of that happening for TV shows, but that doesn’t mean it hasn’t happened.
Unfortunately, there is no “master list” of movies/shows for which Plex had to remove posters.
You can add your own posters for movies and tv shows to the respective media folders. See the Local Media Assets support documents in Plex Docs → Your Media.
To specifically answer your question, the posters downloaded by Plex Media Server are stored in the Metadata folder of the Plex Data Folder. However, neither the files nor the folders use the movie/show name, and the files do not have a .jpg/.png/etc. extension. See the example below.
Be careful modifying ownership/permissions in the Plex Data Folder. If done incorrectly it can cause problems with Plex Media Server.
Also, you may need to refresh metadata after rescanning a movie/show in order for Plex Media Server to retrieve new artwork (if it is available).