If you have requested a W-9 from the vendor and they have refused then you can file a 1099-NEC with the IRS with the word 'refused' entered where the EIN or Social Security number would be.
The IRS requires three requests for a W9 in order to qualify as a refusal. Ideally these should be written so that you can prove that you tried.
Once you have those then you send the 1096 and 1099 to the IRS as well as the 1099 to the contractor just as you would to anyone. Just with that 'refused' written on it.
@jebhawk
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