They had an agreement that upon tax time, the daycare provider would provide them with their tax id (ss#) for claiming what they paid.
Was the agreement oral or written? If it was oral, does your friend have any evidence of this agreement other than his/her own testimony?
When you hire a daycare provider ALWAYS get the tax ID information at the start. That's when the provider will be motivated to give it — if he or she wants the income, he or she needs to provide the information or lose out.
What should my friend do?
Not enough information to answer that question. Possibilities include suing the provider to get this information and/or for damages caused by the loss of the credit and reporting the daycare provider for possible tax fraud to the IRS. Whether a suit to obtain the information can be done in small claims court depends on the small claims court rules in whatever state your friend resides, and you did not indicate the state. Information on making the tax fraud report can be made on the IRS web site here: reporting tax fraud
Is there still a way for my friend to claim this expense?
Internal Revenue Code § 21(e)(9) denies the credit unless the taxpayer provids the name, address, and TIN of the daycare provider. It allows the IRS to make an exception, however, if the taxpayer “exercised due diligence in attempting to provide the information.” So, I suggest your friend request the information IN WRITING and include a Form W-9 and self addressed stamped envelope to make it as easy as possible for her to comply. I suggest your friend remind her of the agreement in the letter and state that your friend may sue to enforce the agreement. Finally, I suggest the letter states that your friend will report the provider to the IRS no matter what happens anyway. Your friend should send that letter twice, once by regular first class mail and once by certified mail, return reciept requested. The former is to ensure she'll see it, the latter is to track her response. Your friend should keep copies of everything, of course. This way your friend can show the diligent efforts he has made to get the information.