Hi! I love the kinds of puzzles you're describing as well! I've designed quite a few myself, and used to participate in puzzle groups and competition sites quite frequently until I got more involved with board games.
When I think of a web-based puzzle competition, I think of something like the World Puzzle Grand Prix, Logic Masters India, and the U.S. Puzzle Competition, among many other national competitions. These all have the user solving on paper, then inputting a string into a rather outdated interface to verify they solved the puzzle correctly (a single row of a Sudoku solution, for example).
I would imagine you're thinking of something with more of a web-based interface, which would be theoretically possible. Unfortunately, I don't think the BGA framework is particularly well suited to this, since it relies heavily on BoardGameGeek game and publisher listings. Making an idea like this successful would likely require involvement from Admins.
Are you imagining puzzles coming from an external source or generated by the server? The puzzle communities I've frequented are quite insistent that hand-designed puzzles are a far more interesting and personal experience than randomly generated puzzles, and I very much agree. All of the puzzles from the competitions I mentioned are hand-designed.
Edit: An additional thought. Logic Masters India's competition interface is based on Flash, which is going to be no longer supported at the end of this year. They have been a prolific source of puzzle competitions for several years, and I'm not aware of any migration plan they may have. That would be a really interesting partnership.
