I’m a chess enthusiast, and I’ve been trying to encourage a Finnish chess forum to switch to DIscourse. So I’ve been looking into this exact question and played around quite recently. My top recommendation is to utilize Discourse’s built-in iframe feature, as someone already mentioned. You just look allowed iframes up in admin settings and whitelist chess.com, lichess.org, maybe some others. They all support iframes on their end. A user can then copy and paste a string like:
iframe src="https://lichess.org/embed/jYSkjcuG?theme=auto&bg=auto" width=600 height=397 frameborder=0></iframe>
A really neat box is created. The box has a chess board and arrows which take you back and forth in the game. If the source has any comments, they get shown, too. If the recipient really needs the PGN file, for example to analyze the game with their engine, they can access the file on the source site. All it takes is to educate your users a little and they’ll love it.