If you have kept no records on child care providers and cannot get their names and SSN's, you can use Form W-10, Dependent Care Provider's Identification and Certification, to request this information from the care provider.
If you are unsuccessful and cannot provide information regarding the care provider, you may still be eligible for the credit if it is shown that you exercised due diligence in attempting to provide the required information. The due diligence steps you took must be listed on the Reverse of Form 2441. One of them should be that the provider refused to provide Form W-10.
https://www.irs.gov/pub/irs-pdf/i2441.pdf at page 3.