Non-residents can get both US corporate persons and US bank accounts, though it takes a bit of work these days. You can form an LLC trivially online (I used LegalZoom for a Nevada LLC, on advice of my accountant). Getting a bank account is modestly more difficult. The easiest way is to go to a multinational bank which has branch offices in your country and get them to pull the strings to open up a US account for you on behalf of your US LLC, with the person having signatory authority being you. This keeps everything nice and tidy with their Know Your Customer processes, most of the time. Citibank is my go-to bank for arranging this for Japanese people, but in principle, you can get it from any bank willing to give it to you. (Word to the wise: they may be suddenly radically more willing to pull the strings if you are vague about your revenue numbers. "We're just establishing the US LLC on a trial basis with one product, but may redirect more of our business through it as time goes on. So, I don't know, call it $X0k per month immediately and potentially higher depending on future conditions.")
Note that international taxation is complicated. Talk to an accountant. Seriously. Talk to an accountant! I probably overpaid at least five figures over the years due to not taking this advice earlier.