Why doesn't Watsi open source its code so developers can donate their time?
That said, take in to consideration that Watsi is quite likely using a bunch of the open-source software you are already familiar with. You can contribute to these projects, and thus indirectly be contributing to Watsi.
What I would like to see is non-profits sharing a list of the open-source software they are using (perhaps even weighted to what piece is most important to their operation) and thus giving eager developers such as yourself a roadmap towards helping them. (Bonus points for ability to reference specific bugs in said software, but that's somewhat rare).