According to the Internal Revenue Service (IRS), the answer is yes. The IRS Hiring Employees guidance reads: "You are required to get each employee's name and Social Security Number (SSN) and to enter them on Form W-2. (This requirement also applies to resident and nonresident alien employees)."
Do not confuse this requirement with your I-9 requirements. You cannot require an employee to produce specific documents for I-9 purposes. But you can ask specifically to see a Social Security card for payroll purposes separately from when the employee completes the I-9 form. If a new hire does not have a social security card, you should direct him or her to the Social Security Administration to get a replacement card or to apply for a new card.
An organization run by AI is not a futuristic concept. Such technology is already a part of many workplaces and will continue to shape the labor market and HR. Here's how employers and employees can successfully manage generative AI and other AI-powered systems.