Your Bio says your title is "consultant." As such, you are probably required to provide the information to the customer's auditor. It depends on what was agreed and the conditions of the contract between you and your customer.
Understand the situation: your customer has a business, and you have crucial information that allows the business to run. If you are unavailable for any reason, the customer needs to be able to continue their business. OTOH, if someone uses the passwords and things go awry, you may be blamed when you had no part in it.
There are a couple of ways to satisfy both parties. Escrow agents are one way - you give information to a neutral third party, who holds it against the eventuality of your unexpected departure. If you default, the information is turned over to the customer. Escrow agents charge a fee for their services.
Another way is a sealed-envelope system: you turn the passwords over to the auditor sealed in an envelope, or in nested envelopes. If the seal is broken, you cannot be held responsible for the changes to the network. That way, the customer can access the passwords in an emergency, but you are also protected. If something happens and you are blamed, ask to see the sealed envelopes.
But this case sounds like the "Golden Rule" - "the one with the Gold makes the Rules."
Good luck.