Trevor the Guru here!
To use Shopify as a whole, you wouldn't require an SSN, however, if you're wanting to use Shopify Payments as a credit card processor we would require the last 4 digits of your SSN. The SSN is required and is outlined in our Shopify Payments Terms Of Service: Section B-3.
Shopify Payments is only one of many payment processors, however! We have a huge list of available payment gateways for the United States here as well as a list of which payment processors are direct or external here.
Feel free to reply here, or phone/chat in if you have any other questions!
All the best,