Long, long ago, early in my career, I spent about fifteen years in the non-profit sector.
You don't ignore office politics, but you don't take sides either unless there is a crisis brewing -- something illegal, highly unethical, or financially dangerous. When you work in IT, you're in a "support" position, rather than a "line" position. Your job is to support. So when there's a big pissing match between two line functions, your job is to support *both* sides.
Often this means documenting business processes that sort of evolved via the lava flow antipattern; 50ish is the size where things start to get out of hand, because it's the size where the amateurishly hacked-together processes that keep the organization running start to break down because everyone can't be aware of everything that's going on in detail, in real-time. Make it your business to understand what business systems (not necessarily computer systems) *accomplish*. That puts you in a position to offer a third way, the one that emerges as obvious to everyone once somebody has figured out what's actually going on.
It's supposedly hard to implement changes in non-profits because of the consensus-driven decision making processes, but I found that I could make that process work for me. Lack of understanding is a vacuum; presented with a clear picture people usually line up behind the obvious solution quickly. But you do have to do your homework. Never surprise anyone with anything in a meeting. Bring people up to speed with things you're going to say about their work *before* the meeting so they don't feel blind-sided.
In a crisis be prepared to do the right thing. If you're in a non-profit they're paying you below market rates, so you can do better elsewhere. There is no call for getting yourself sucked into something that offends your self-respect. I resigned one job because my superior (the COO) was doing things that were financially reckless and improper (spending without proper authorization). I informed the CEO in my exit interview. That was my solution to the problem of not getting drawn into a persistent pattern of dysfunction.
When you handle sensitive information, just ask yourself what is the professional thing to do? Be discreet. Resist the temptation to peek at data, and when you *do* accidentally learn something you're not supposed to know, disclose that to the responsible parties. Be trustworthy, and present a trustworthy face.
Finally, don't let them pay you far below the market rate for your services, and expect a really good benefits package, including 1.5x to 2x the vacation you'd get in a for-profit. Insist on the respect due a professional. Non-profits are full of young people who haven't learned that the IT guy isn't there to be kicked around when they're frustrated, and the fact that you're in a support position rather than a more glamorous line position doesn't make your work any less important.