How do US companies pay for services (SaaS)?

Different companies, different ways. And it also depends on the size of the expenditure. A small company will often simply use credit cards and then account for items that way. A larger company with better fiscal controls will probably have a much tighter voucher / approval / direct deposit mechanism for purchases over a certain dollar amount.

As for Poland versus the US – again, depending on the service or product – some purchases are required (or have a preference) to work with US companies, if for no other reason than problem solving – if they need service, it will be in English on the US clock, not on Central European time. And then there’s the legal side – which country’s laws govern the transaction? US or Poland? So there is a preference for a US provider.

Some US companies set up payments through services of a (US) bank, called “merchant banking” and related companies that work with them to provide the needed software and services.