Alex here from the Shopify support team.
To use Shopify Payments, an SSN is required and cannot generally be replaced with a different tax ID. You can reach out to our support team directly to discuss your specific instance, just in case an exception can be made.
If you're not able to use Shopify Payments, you should be able to use a third-party gateway to process payments - you can find the list of gateways available in the USA here.
I hope this helps clarify things a bit! Please feel free to follow up on this thread with any further questions, I'll keep an eye out.
Alex | Support