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Where is the employee working? In NJ? Or NY? Or are you doing courtesy withholding? If they are working in NJ, they are still taxable there.
If you are doing NY SIT WH, then you most likely are registered with NY. The Yonkers local tax is sent in with the NY SIT payments (paid to the same agency, on the same return). When the employee filled out their IT-2104 for their NY withholding, there is a section where they indicate they are a Yonkers resident. There is no separate Yonkers registration needed.