I've recently had good experience working with Selenium to create automated acceptance tests for a project quite similar to Moodle. I'd say Selenium is the right tool for this job and is quite mature as well.
However, the tool is not the important thing, it's if the application is designed to support automated testing in the first place. I don't have any experience with Moodle just yet, so I can't say if this will be easy.
Automated testing of Moodle is definitely something I will look into in the near future. If you (or someone else) is interested, I'd love to collaborate on this.