Foreign student opening U.S. business
You could certainly register an LLC or a Corporation as a non-U.S. resident, without having a U.S. address or social security number.
You may technically register in any state. Typically, Wyoming or Delaware are the top recommended states due tot heir business friendly laws. We have a research article titled 'U.S. Companies for Foreigners' which will compare the top recommended states and guide you through the online application process.
Keep in mind while you may register your business in any state, if you plan on having a physical presence at some point, such as a warehouse, storage/shipping facility, etc., your company would need to be registered in that particular state.