The main way to collaborate on someone's book is to become a collaborator of the book, or a member of the book's organization. But sometimes you simply want to submit a small contribution, without becoming a regular collaborator.
On GitHub, one can use the popular Pull Request feature to do that. To do so, you would clone the original work, make changes on your own copy, and submit back your changes by submitting a Pull Request to the original author. Then the author is free to integrate your changes.
There is no such feature on GitBook.com yet. For now however, you can still link your book to GitHub and host it there. After that, people will be able to open Pull Requests your GitHub repository.