In your situation, you cannot rely on the local webstorage, but it should be possible to "serialise" the important states and data of your game session and store the info in a shared location (remote database, etc.).
Possibly doable, but certainly not trivial, and probably a sizeable amount of work to get to work right.
Maybe you can design your game around this limitation, and make the "save" data minimal and easy to manage
In the current state of things, and given the philosophy of the framework, I don't think there's an easy solution to networking multiplayer saves. Maybe someone will prove me wrong, but it sounds like you'd need at least a database, some server logic layers with php/ajax, some custom load/save mechanism, etc. It would be convoluted, and you would lose the benefits of using C2, i.e. the high productivity for simple games.