Running an online business from Canada
Technically, you do not need to form an American company to continue doing your business. Although you should register your Canadian company as a foreign entity in the state where you do most of your business in the U.S. I would definitely talk to an American accountant about this as well, as I do not know the tax implications of a Canadian company doing business in the U.S.
It does make sense for you to register a U.S. company, though, an LLC or an S-Corporation. This will give your customers a piece of mind that they are dealing with a domestic company, and will also make the tax situation a little simple, I believe. You do not need to have a social security number to form a company in the U.S. and you can even have it wholly owned by your Canadian corporation.
As for the bank account, U.S. companies typically have a bank account to deal with the everyday payments and invoices, but if PayPal is sufficient for you, then there is no actual requirement for your company to have an account at a bank.