A US Social Security Number (SSN) is not required to start working in the United States on your TN Visa, but you might still want to get one.
Neither the USMCA (NAFTA) nor USCIS defines how a TN Visa holder must be paid. They simply state that there must be employment or contractual arrangements between a foreign national and a US entity. Only self-employment through a US company is barred. We discuss compensation and self-employment on a TN visa here.