Live in IL, registering a new business
In the United States you are not limited to registering a company to match your state of residence. The limitations occur in the location and type of business you are conducting.
If you are operating a business solely online that carries it's business through a website, then yes, you would be able to register in another state such as Louisiana, Wyoming or Delaware.
If your business would have a physical presence (this includes conducting your business in person, having an office, employees or physical property to be owned by the business) in the state of Illinois, then you would have to register your company in that state, even if originally it was registered in another state.