Another place could be local libraries' digital collections. Many libraries offer free e - books and audiobooks for borrowing, and they may have cowboy short stories. You can check their websites and see if they have a search function specifically for short stories in the western or cowboy genre.
Some websites dedicated to short stories might also have free cowboy short stories. For example, ShortStories101. They often categorize stories by genre, so you can easily navigate to the cowboy section if available.