Thank you for posting your concern in the Community. It's certainly strange that nothing happens after entering the login credentials. I'm wondering if you are using a browser to access MyRogers since you have posted in this thread. Typically, if you were to use a browser on your iPad to access MyRogers, you should get redirected to the App Store to download the app.
If the redirect is not happening; you can visit the App Store and download the MyRogers app and share your experience with us.