Unfortunately, you will not qualify for the American Opportunity Credit without that federal ID number.
According to the IRS:
Educational institution's EIN is required. To claim the American opportunity credit, you must provide the educational institution's employer identification number (EIN) on your Form 8863.
There is a new requirement for 2017 that requires a Federal EIN in order to claim the American Opportunity Credit. Without this number, you will not be able to claim the AOC but could still be eligible for other educational benefits such as the Lifetime Learning Credit.
More info on the Lifetime Learning Credit:
https://ttlc.intuit.com/replies/5358151
