I’ve used Blackboard, Moodle and eCollege and none of them offer integration with Google Apps in the way you frame it.
We have a custom single sign-on that provides access to both Google Apps and the LMS but the Google Apps (for example Docs) do not work within the LMS (users are redirected outside the LMS).
Yes, you can Integrate LMS to Google App, and one very good option for such LMS is Paradiso LMS. Leverage and share work files, documents and online training content across Google Apps with Paradiso Learning Management System. Some of its features are:
1. Google Apps LMS Single Sign-On
2. Enhanced Collaboration with Google Apps LMS Integration
3. Importing Files into LMS from Google Apps
4. User Auto-Enrollment
These features are a great option in an eLearning process. Most of the good LMS companies understood this time ago, due to the easy way to work with the google apps as words, sheets or slides. Another excellent move of using it is the size of the cloud source, where you can place information as you wish.
Even knowing this, many learning platforms still having troubles implementing this important features. The Google app as an integration of the platform is an excellent complement and, I would say, necessary for the correct handling of an eLearning system.
By the way, this kind of integrations or apps as you said are necessary when you are trying to integrate your LMS into a mobile app. As you can see on the following blog, “eLearning app”, the integration with tools as Google is necessary to provide a fully mobile experience.
Finally, answering your question, the answer would be: depends on the LMS. Depending on the platform you choose you will be able to integrate this necessary feature or not. To have more information I suggest you read the following site.