We were given ThinkerShields for our class and I am trying to use this with my Chromebook as that is what all my students have. In order for my students to use the ThinkerShields will I need to get a subscription to the App and pay a monthly fee? This doesn't seem right as it would be free if I was using Microsoft?
The ThinkerShield has absolutely nothing to do with it. This is about the ability to use the Arduino Web Editor on a Chromebook. That would be the case with just the Arduino board alone, whether or not you plug a ThinkerShield into it.
If you want to use the Arduino Web Editor with Chrome OS you will indeed need to pay for a monthly subscription to the app.
There is a way to use the standard Arduino IDE on the Chromebook, and this will be completely free and more powerful than the Arduino Web Editor: