This is from the IRS.gov website about your situation:
My babysitter refused to provide me with her social security number. Can I still claim the amount I paid to the babysitter for child care while I worked? If so, how do I claim these child care expenses on my tax return?
Yes, if you meet the other requirements to claim the child and dependent care credit, but are missing the social security number or other taxpayer identification required ID number of a provider, you can still claim the credit by demonstrating "due diligence" in attempting to secure this information.
If a provider of child care refuses to give the identifying information, the taxpayer can still claim the credit however, the taxpayer must provide whatever information is available about the provider (such as name and address) on the form used to claim the credit Form 2441 (PDF), Child and Dependent Care Expenses, or Form 1040A, Schedule 2 (PDF), Child and Dependent Care Expenses for Form 1040A Filers). The taxpayer should write "see page 2" in the columns requesting the missing information. The taxpayer should write at the bottom of page 2 that the provider refused to give the requested information. This statement will show that the taxpayer used due diligence in trying to secure and furnish the identifying information.
For more information refer to Form W-10 (PDF), Dependent Care Provider's Identification and Certification, the instruction for Form 2441 (PDF), Child and Dependent Care Expenses, Publication 503, Child and Dependent Care Expense, or Tax Topic 602, Child and Dependent Care Credit.