This article shows you how to check on the status of an app in the Apple App Store. Is it possible for you, as Developer, to log in and check the status? Yes, you can, but it might not be necessary. Whenever the status changes, the Developer of the app will be notified via email.
Here is an overview of the status an app can have within iTunes Connect:
A change in status will be accompanied by a comment if needed. For instance when an app is rejected, then the reason why will be added.