I haven't yet tried local storage, but considering this line from the docs "accesses the current domain's local Storage", I worry it doesn't work after "updating" a game (deleting an old file, uploading a new file to a new, randomized directory).
Please correct me if I'm wrong, but wouldn't an update button on an existing file in the dashboard solve this problem? Instead of only new files, or deleting existing files, what about "overwriting" existing files? Keep the old directory the file lives in, delete the contents, and then upload new contents to said directory. This way concepts like PlayerPrefs in Unity's WebGL builds would work.
I'll try interfacing with local storage from a Unity WebGL build later and update this post if it works for me.