Yes, you can 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 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 use of due diligence in trying to secure the identifying information for the claim.
You will not be able to e-file.
Well you could paper file the return and include a statement explaining why you could not get the SS# and all the efforts you took to get it BUT the IRS may not allow the explanation or deduction.