GitHub is not intended for hosting active websites that require a database. GitHub is intended for hosting code repositories, such as the actual source code of Moodle as it is under development.
I'm not even sure how a system like Moodle that requires a database could run directly off a GitHub repository. GitHub Pages runs off repositories, and can be used for hosting static sites. See http://blog.teamtreehouse.com/using-github-pages-to-host-your-website. However, the security implications for hosting a static site are quite different, of course.
Now if your organization wanted to use GitHub to store the codebase from which they deployed their Moodle code, that would be fine - as long as they excluded from the repository things like the data directory (which shouldn't be in the document root anyway), and the config.php. If you were concerned about private themes or plugins being cloned by other GitHub users, though, you would need to have a private GitHub repository, which you can purchase for a fee.