Assuming you have already requested this information from your dependent care provider, and they still refuse to provide it. here is the procedure that you must follow, in order to claim the dependent care credit without the provider’s SSN or EIN.
First, understand that you can not e-file your return without the caregiver’s SSN or EIN. You will have to print, sign and mail it to the IRS. There is no alternative or workaround for this.
Next, print out a form W-10 (Dependent Care Provider’s Identification and Certification) from the IRS website at http://www.irs.gov/pub/irs-pdf/fw10.pdf.
Fill out the W-10 form with all the information that you know.
Make 2 additional copies of the filled out W-10 for your own records.
Mail the form to the caregiver via certified mail. Don’t lose your post office receipt.
The caregiver has 10 calendar days to respond. If they respond with the information you need, then you’re good to go. If they have not responded, then on the 11th day after you mailed it certified mail:
Select the “print and file” option for your tax return. This will print only the documents required for filing, along with an instruction sheet that contains the address of the specific IRS processing center you will mail it to.
Include in your mailing a copy (not an original) of your proof of notification (the certified mail receipt you got at the post office), and a copy of the W-10 that you sent to the caregiver.
That’s it. While this will delay your refund (it’ll be 6-8 weeks before you get it), you’ll still get the credit. At this point, it’s no longer your problem – it’s the caregiver’s. Per @Carl