Anything except, “I’m sorry!”
You can say, “I apologize,” but that’s not what the customer is looking for. You can begin to tell your story about what happened, but that’s not what the customer is looking for.
The best way to apologize is to let the customer vent first. Don’t interrupt, just take notes and make empathetic noises. You can even tell the customer that it makes you mad too. Second, ask the customer what their speed of need is. Do they need it by tomorrow? Do they need it today? Or did they need it yesterday?
Most customers will have needed it by yesterday. This is your big chance to be memorable, by getting it there the day before yesterday (just kidding). The reality is their need for speed will determine your action plan for recovery. Realize that you have hit a flash-point and are in jeopardy of losing the customer. Therefore, any action you take towards recovery is a positive one.
The interesting news is that most big companies have firm policies in place that preclude memorable recovery: needing an invoice, needing a customer number, needing a return shipment authorization, and other crapola that no angry customer wants to hear.
FINAL ANSWER: Tell them what they want to hear. That you apologize, that you understand how they feel, that you are meeting with the appropriate people to get a resolve, and that it will be done in 24-hours. No blame, no excuses, no drama.
EPILOG: Follow up with a personal call and a personal note of thanks. This makes the recovery complete, and paves the way for the next order, or a favorable referral.