Cocoon definitely does support WebStorage, so it's probably something to do with your implementation. I'm using WS for persistent stats in a game I'm currently working on and even with the current, rather buggy beta release, it's behaving perfectly.
Otherwise, you can set up your own online score table with PHP and MySQL by following ArcadeEd's excellent tutorial. Obviously this requires a data connection. You should probably follow that tutorial even if you have an offline score system, as it deals with working with arrays.