Selling a product brand from Canada in a US state.
It sounds like opening a business (either a corporation or an LLC) in the state of Illinois would be the best way to go about your business plan to be able to distribute your products in the U.S. without having to ship back and forth between countries.
You can open a U.S. company without being a U.S. resident, so this shouldn't be a problem for you. Once the business is registered with the state, then the company will be able to apply for an EIN, and the state sales tax id, which is a permit that allows you to charge your customers state sales tax, which in most states (including Illinois) is required when selling products to consumers in that state.
Typically speaking, I would suggest registering an LLC, as this entity type would require less maintenance and offer a little more flexibility, especially if you will have someone else managing and operating the U.S. branch.