Passwordstate gives you the possibility to request an access to a password or passwordlist by clicking on the passwords menu (the first icon) on the lefthand side. Through this option you can request an approval from admins of the Passwordlist/Password. Permissions can be get only for a certain time window or can need a handshake approval (two person).
There's also an other feature which could be interested for you: When you have an account properly setup with password rest options you can use the checkin/checkout feature. This will show a Password only after it has been checked out by the user. After the work has been done, the password needs to be checked in again. Then the password reset comes into play, it will immediately generate a new password and set this new password on the related system.
So no one (expect passwordstate) will ever now the current password and the user can't write it down.