Is there a way to use an API call, webhook, action, whatever...WHILE in EDIT MODE

This looks like it can be done with a form rule.

First make a formula field to extract the employee number out of the record and have it be in a stand lone formula field that you put on the form.

Then make a form rule that says

When multiple conditions are true.
when the [EE number formula] is not (blank)
the [Employee Number] is blank

Change the [Employee Number] to the value in the field  [ Employee Number formula]

Make sure to uncheck the box at the bottom to make the rule fire all the time.