Unless I'm missing something, I don't see a way to get an FEN string for a vote chess game unless you make it yourself.  Could you add to vote chess the easy "fen string" acquisition like in regular games so that people could make diagrams for study/discussion easier?

isn't there one at the bottom of the details pane?

I've never seen one there on any vote chess game I've played.

It's not possible as far as I know however copy-paste the move list as a PGN file works a treat. Just set the start point at the end of the moves.