It is possible to claim the American Opportunity Credit without an SSN, but you will need a verifiable identification number allowed by the IRS. 

According to the IRS American Opportunity Tax Credit page,

You may not claim the AOTC unless you, your spouse (if you are filing a joint return) and the qualifying student have a valid taxpayer identification number (TIN) issued or applied for on or before the due date of the  return (including extensions). A TIN is a Social Security number, an individual taxpayer identification number (ITIN) or an adoption taxpayer identification number (ATIN).

You may not claim the AOTC on a later original return or an amended return if the TIN is issued on or applied for after the due date of the return (including extensions).