Is there an overhead agency that I can contact?
Do you know whether the bank is not in existence because it was acquired, or because it became insolvent? Often times when a bank ceases to exist, even in a distress situation, the deposits are moved to a healthy successor bank. If you feel comfortable disclosing the name of the bank, it may be possible to see where the deposits ended up and you could call that entity directly. Depending on how many years it has been, the account may have been reported to the state as "unclaimed property" in which case you would have to deal with the agency in that state.
Hi: I would contact the FDIC. They should be able to help you
Every state has an "Escheated Property" department. In many cases, you can prove your claim to the value of the account and get all or part of it back. You will not get any interest earned. Also, be aware that these departments are very thinly staffed so expect to be directed to websites quite a bit and do a lot of legwork yourself. The Secretary of States' Office could also be a resource. As Barry mentioned, the FDIC may need to be contacted depending on how long the account went unclaimed before the bank was sold or acquired.