"Can I get the credit."
Maybe. You should have collected their SSN on a W-10 before you hired them. If they provide the SSN now, you can claim the credit. If they refuse, you can file by mail and attach a written statement. You must explain your diligent efforts to obtain the provider's SSN, and you must give as much identifying information about the provider as you can (name, old and new address, phone number, etc.). This IRS will review your statement on a case by case basis and may award the credit, but it is not guaranteed.
Turbotax might have a check box for "provider refused" that will allow you to finish and print your return. (I haven't tested it this year.). If not, use a fake SSN and then cover it up with white-out on the printed copy, write "refused" and mail your return with your statement.
