Students can write an embedded Google Docs that are automatically saved to their Google drive as long as their teacher has given them access in the assignment. Please see the article Can my students write in a Google Doc? for instructions on how you can give students access to an embedded Google Doc.
Once that is done, all students need to do is log into the app and click on the assignment and begin writing. In the writing screen, they will need to click inside the Google Doc. From there, they will also have access to the Google Docs toolbar.