If you're going to give it a try, I would recommend giving fuzzing a shot; it's a very effective way to find interesting and potentially exploitable bugs. I'm not too familiar with the tools these days, so I don't know if there are fuzzers you can just download and start messing around with, or if you still need to roll your own to effectively target the full attack surface of the application you're interested in, but I imagine there are plenty of resources on the subject online.