I haven't actually used webstorage that much, but as I remember there is something about if you clear the cache you might loose it, but you can check that in the manual, as im not sure :D
But yes you could do that.
Just add this to the event where you want to save the score.
Webstorage -> set local key "Highscore" to Highscore
System -> on start of layout -> set Highscore = Webstorage.localvalue("Highscore")