There are many varieties of "offshore" for this question. I am not routinely inconvenienced by being an American living in Japan with regards to charging for things. If you're a Nigerian living in Nigeria... things are going to be a wee bit more difficult.
One does not practically or legally need a physical or corporate presence in the US to charge cards. Of the available options, I use Paypal to charge credit cards -- they had the least involved process to get a Paypal account (Paypal Website Payments standard + Paypal premier account) years ago, and I bootstrapped that into their I-can't-believe-its-not-a-merchant-account (Paypal website payments pro + business account) after a few years of doing (relatively) significant volumes through Paypal proper. That has been impressively painless.
Privilege escalation is, by and large, the easiest way to get into the US banking system. As soon as you're in a bank's systems and they have their Know Your Customer checkbox ticked next to your application, your odds of not getting held up by fraud review for the next product you add go up greatly. (This is why the tech incubator I used to work at would set up anyone going to the US with personal checking/investment accounts at US multinational financial firms. As soon as you have a foot in the door, everything else gets easier.)
Understand that businesses do not refuse to do business with foreigners because they enjoy screwing with you -- it is strictly risk-mitigation. There are a variety of things you can do to reduce perceived risk. In general, things which tend to indicate you're a "real" business tend to work. (Tax forms are practically magic spells -- they look official, it is virtually impossible to second-guess their contents, and the IRS will virtually never tell you "Piss off, we don't want your money.")