Website company closed down, but he's hosted somewhere, that's easy. Contact the hosting company, determine who's being billed for hosting, if it's your client a 5 minute conference call with verification of his credit card on file they let the hosting know you have rights to access by FTP. Done.

Typically it should / would end there UNLESS the site was dynamic - a WordPress, Joomla, Concrete, etomotie, etc site then it get's into issues of accounts and permissions.

Before you think about access think about liability - do you really want DIRECT access? do you want the responsibility of their user name, password and if it's breaks you're the one they blame?

Insurance and contracts

Are you insured if your access accidentally deletes everything? Does your contract state a protection of arbitration if they make claims you deleted, took, added or changed their website?

If the website was a room in strangers house and a pile of cash was left on the table, would you know their character well enough that wouldn't accuse you of theft? Having direct access goes from light dating to marriage, are you ready for commitment?

Before you do anything make a back-up Whttrack is a great program to make a static web backup. Make it for yourself first and foremost because you never know what will happen.

Back up the database, files, images, structure, accounts...everything!