Nothing like referring to the authentic source, straight from the horses mouth (IRS). It's an easy 1 para read:
Question
My babysitter refused to provide me with her social security number. May I still claim the amount I paid her for childcare while I worked? If so, how do I claim these childcare expenses on my tax return?
Answer
Yes, you may still claim the child and dependent care credit when you're missing the provider's social security number or other taxpayer identification number by demonstrating due diligence in attempting to secure this information.
Claim the childcare expenses on Form 2441, Child and Dependent Care Expenses and provide the care provider's information you have available (such as name and address). Write "See Attached Statement" in the columns missing information. Explain on the attached statement that you requested the provider's identifying number, but the provider didn't give it to you. This statement supports the exercise of due diligence in trying to secure the identifying information for the claim.
