Immigration law does not permit anyone other than a SSN holder to work. It does not matter whether the person is employed, self-employed or a volunteer. My niece was most upset that she could not do voluntary work whilst waiting for her work permit.
I agree with sjr. I can’t imagine the IRS would be fussy. An ITIN, rather than a false SSN, might indicate that the worker is seeking to become legal. I did have a client years ago that employed several ITIN holders and people with fake SSNs. He got a call from NC employment security commission to say that he had one quarter to get them legal or off the payroll. Not sure if the state had that authority, but it focused his and his employees’ attention.