There are three ways that an ISPConfig email user can use to change his email password (when he has no client login).
Use the Mail user login in ISPConfig
Email users can login to ISPConfig with their email address and mailbox password by using the normal ISPConfig GUI login. They will get logged into a reduced ISPConfig login which just allows them to edit settings of their own mailbox.
There is a plugin for RoundCube available that connects to ISPConfig via ISPconfig remote API. This enabled Email users to change their passwords within RoundCube.
Plugin for RoundCube:
There exists a plugin for SquirrelMail as well which allows mailbox owners to change their passwords within the SquirrelMail webmail client.
Plugin for Squirrelmail:
Tutorial about installing the SquirrelMail plugin: