Here's a bit crazy idea, but one that would fulfill the requirement:
Create a simple (but secure!) system that holds all the passwords for different services. Give everyone individual access to that system. The system provides the password for the service the user requests to access. Whenever a user requests password to a site, for example eBay, that request is logged. The user is given the current eBay password, and a randomly generated new password. The user then must immediately change the eBay password to the new one, whether or not she decides to use the service.
This way the user A is accountable for anything done in the period starting from her getting the password, and ending in a new password request. When user B requests a password, she gets the same password user A knows, but B then becomes accountable and thus must immediately change the password in order to deny user A access.
The full history of passwords should be stored for admins to gain access to the service in case some user does not change the password.
This is no way bulletproof, of course. There is a brief period of time before password change where user A could access the service while B is actually accountable. Also, if B requests a password, does something foul and afterwards changes the password to something completely else she could plausibly claim A changed the password, and the one B got never worked in the first place. But I don't think this sort of activity a was a concern in your case.