Yes, you should still file Form 1099-MISC, even if you do not have the Social Security number.
The IRS may impose penalties (which can be substantial) on you if you do not file these forms for all applicable contractors by January 31, 2017.
Please see below for steps you will need to take to try to obtain the contract employee's social security number. Please note that you will probably have to file your 1099s and 1096 using paper forms if you are unable to get this information.
