This relates to the rule, 

"To qualify for the EITC, you (and your spouse if filing a joint return) must have the required Social Security number (SSN) issued by the Social Security Administration on or before the due date of the tax return (including extensions)."

Did they use an ITIN to file the extension request?  Is that where IRS is looking for it?  Did they keep a copy of the extension request, or did the preparer?  I would start by asking for help from the Local Taxpayer Advocate.