If you don't use ASP.NET WebForms, you won't use ViewState.
ViewState is a concept with very bad reputation, you have to avoid it.. actually. I heard that ASP.NET WebForms 4 has a lot of improvements there, but unfortunatelly without details. But you are not able to just 'say' use localStorage instead of ViewState, since WebForms controls are dependend on it.
So, you if you wish to manage client side state with localStorage you have to create own code. And you obviously need to pick up another technology. If you are on ASP.NET stack - ASP.NET MVC is best choice.