My employer has filed a PERM application for my GC and shared the Case number with me.
Is there a way to track the progress of this application?
You should be able to track your case by using PERM tracking number.
Once Company file Perm/Labour they get a unique ID. To check status of the application you need to visit this site: http://www.plc.doleta.gov.
However, you cannot create or register your self into http://www.plc.doleta.gov . You need to ask your company HR to create a sub account and give you access.