It's been requested several times before that we support some sort of persistent storage for scenarios on the Greenfoot website. This could be used for high-score tables, or "save games" (remembering which level you'd reached in a game) or other uses. I'm glad to say that this functionality will be added to the next Greenfoot release, 2.2.0, which is coming soon (that probably means in about a month!).
Prior to the release, you can help us test this functionality and iron out bugs. I've uploaded a scenario that uses the new system to have global high-scores. As long as you are signed in to your Greenfoot website account (i.e. your profile picture is visible top-right) when you load the page, and are not behind a proxy or over-zealous firewall, you should be able to play the game and set a high score, which will be remembered thereafter. Please let me know if you see anything wrong (e.g. score not saved, or the ranking of the high-scores is wrong). Once Greenfoot 2.2.0 is out, I'll upload the source for the scenario and you can copy the scores aspect, but there's no point for now because it wouldn't work in Greenfoot 2.1.2 anyway.