I can’t log out of an account because we don’t know password. When I try to reset password, Apple says the account doesn’t exist.

If necessary, contact Apple ID Support for assistance.  See the Apple Support articles below for contact information.  

Apple ID - Official Apple Support.   -

Contact Apple for help with Apple ID account security - Apple Support.  *   (Link doesn’t always load)  

Get Support      -    Work through the screens to get a chat option. You may have to pick a slightly different problem.

Apple Support schedule phone call.   -

Apple on Twitter       

From the Apple Support article below. 

Without proof of your identity via security questions and other carefully selected criteria, Apple Support can't help you reset a password or perform any other actions on your account. These policies are audited and reviewed on a regular basis.

Security and your Apple ID - Apple Support.      *

Have you tried Apple's iForgot service?

If you have forgotten your Apple ID password – Apple Support (UK)

Also, I had this with an old apple id. I ended up totally reinstalling and erasing the Mac I was doing this on (my dads old mac Mac without activation lock)