A lookup field in a custom object needs to be updated after the record is submitted for approval and locked.
Is the trigger able to achieve this, even if the record is locked?
The answer is, it depends.
If you have your code written directly in the trigger or in a class designated
without sharing, then the apex code can update records that are locked, regardless of the user who caused the trigger to fire. I tested by creating a trigger on Contact that updates a field on the Contact's Account. When the Account is locked, I can edit the contact record with a non admin user and the field on the Account gets updated.
If you use code in a class marked
with sharing that is called by the trigger, then you will get an exception. If the trigger was executed by an Administrator, then the code will always run.
It is if the user is ;)
If the action is initiated by System Administrator - sure. Otherwise - tough luck, exception will be thrown. It also depends if you've configured the approval process to "Administrator or current approver can edit the records".