Delta Customers Lost Time, Money. How To File Claims, Complaints. Customers of Delta Air Lines may submit claims or complaints for reimbursement if they experience delays, cancellations, or luggage problems. Visit Delta’s website and fill out the online form there, including details about the incident and any relevant receipts. Report lost or damaged luggage online or at the airport within a day of the incident. Additionally, customers can contact Delta’s customer support by phone or via social media. If first attempts at resolution are insufficient, take the issue further by filing official written complaints or contacting regulatory bodies for additional help.