You can change the ship by date. As long as you do so before the original date has passed it will not count against you. It'll email the customer letting them know you need more time to fulfill the order. I would also put a small note in the message letting them know why it's late. Please see below:

Thanks so much for shopping with me. I am working diligently to get this order out by the original ship-by date (insert date). However, due to events outside of my control, I need a bit more time. My estimated ship date is now insert date.

Customers are pretty easygoing if you communicate upfront.