According to the IRS instructions, you should document the attempts you made to obtain the ID number, mail the 1099, file the 1096 with no number and provide the IRS a summary of your attempts to obtain the ID number. You will receive a letter stating the form was not properly completed, but there will be no fine since the form was submitted on time and attempts were made to obtain the information.
I would suggest you provide the non-compliant independent contractor a copy of your intended filing with the IRS (if time allows). In your letter state the documents provided will be your official filing with the IRS due to his/her non-compliancy and the income is still reportable taxable income regardless of the fact the 1099M is incomplete. I would suggest you batch these/this 1099M in a separate batch and not with your other properly completed 1099M.
On a side note, a contractor should not have been paid without, at least, obtaining a completed Form W-9, which should be a requirement as listed in your independent contractor agreement.
For general information on filing Form 1099M, 1096, etc. see https://www.irs.gov/pub/irs-pdf/i1099gi.pdf
Ask The Taxman: When Should I and When Do I Have To File A Return?