This is absolutely possible.
I've setup companies in other countries remotely and I've setup companies in the US in person and remotely. All you need to do to register a company is to file the proper paperwork. If you want to create a LLC or other limited structure then just draw up your articles of incorporation and send them off. The issue is that most states (perhaps all) require a registered agent who has a physical address in that state. I believe there are firms which will act as your registered agent for you. This is so the state can send you official notices and things like that. Fail to file a simple annual report with your state or registration and they'll shut down your company. So, its important but trivial stuff.
If you have a little money (should not be much, maybe $1,000 - $2,000) hire an attorney to do this for you. Most of them have the process quite automated.
You might need to give a power of attorney but better to give this to a licensed profession rather than someone who has nothing to lose from abusing the power you are giving them.