I am based in Canada and ended up opening a US bank account with Harris Bank in Chicago without physically visiting their office. This was about 4 years ago. So maybe contact them and see if they could help you.

You could hire a CFO in the states and have the board authorise them for banking. Then they could open the account. Many companies get their banking done this way in the US (think of Multinationals).

It is fairly difficult to open a U.S. bank account online if you are not a U.S. citizen. U.S. citizens are able to open accounts online because they have a social security number (SSN) to verify their identify. For non-U.S. residents without a SSN, you'll generally have to physically visit a branch location in the U.S. to open the account. The bank representative will examine your documents (passport, drivers license, LLC formation documents, etc) necessary to open the account. In recent years, however, there are more online banking options where a non U.S. resident can open an account remotely. Some of these banking providers include Transferwise, Mercury.co and Payoneer.