They have an
FAQ on the ESTA website:
Question 20:
I Can't Find (or didn't print out) My ESTA Application Number Question: Can I update my application, get my status, or board a plane if I don't have my authorization number?
Answer:Due to security/privacy concerns, we are unable to provide application numbers to people who have lost or misplaced them. If you lost, misplaced, or forgot your application number you will have to re-apply for in order to get another one. Your old application number will be automatically cancelled and a new one will be generated for you.
If you lost your application number, you will not be able to update your contact or travel information in the system. That number is required to locate your file. But remember, it is not required for you to update your travel information in ESTA. Providing that information to us is optional. So if you have lost your application number and cannot update that information the next time you travel to the U.S., don't worry about it.
You are not required to present a printout of your application number to board your flight. That number is transmitted electronically to the airlines.
If you are waiting for your approval status to be confirmed, and you did not keep a copy of your application number to log in to see the status, you will have to re-apply. We do not notify applicants of their status (approved or denied) by email.
If you are unable to re-apply, please come back to this FAQ and click on the "Ask a Question" tab at the top of this page, and send us the exact error message. If you do not have an email address for us to respond to you, please use esta@cbp.dhs.gov. Be sure to give us a phone number where we can contact you to resolve the problem