Assuming you can't convince the check-in agent that you are under 18, I believe the Delta policy is either one valid government-issued photo ID or two forms of non-photographic identification, one of which must be government-issued. So a social security card would work with a second form of ID.

This of course assumes you have an agent who knows Delta policies. I just looked quickly on the website and could only find it under Delta Vacations but with more time to search you might find a better citation:

Travel Identification
• Domestic Travel: All passengers 18 years of age or older are required to present 1 form of government- issued photo ID (such as a valid driver's license) or 2 forms of non-photo identification, 1 of which must be a state or federal agency-issued (such as a U.S. Social Security card), before boarding a flight. Failure to present positive identification can result in denied boarding and no refunds will be granted.