You should never do business with a sub unless you have a filled out and signed form w-9. signed under penalties of perjury. The IRS requirement is to have that w-9 before the end of the year in which payments were made.
Do the 1096/1099 manually and put refused to provide in the tax id number block.
I would call the contractor and tell him you are going to do that, unless he gives you his number. That will result in the IRS searching him out, requiring him to be on back up withholding and issuing a $50 fine.