How would you go about doing that? do you have to write a pitch the same way you would have to for paying markets?
Super easy. Just make a list of charities/non-profits you'd be interested in working with, go online and find their PR/media/communications manager. Just send them an email saying who you are, what you're doing and that you'd like to offer them some pro bono copywriting. Give some examples of the sort of thing you could do, and ask them
Here's a next step that you could think about for once you've got a portfolio together. My biggest hurdle was finding clients, and particularly clients who even considered copywriting to be something worth paying for. But then, on the advice of my graphic designer girlfriend, I set up a business relationship with a graphic designer who builds websites for small businesses. Basically, when one of his clients is getting their website developed, he sells me as his "copywriting expert" who will write everything on their website. Because its already going through him, I never have to talk to clients directly or find them myself. He gets me to quote on a job, then charges that to the client (presumably adding a bit on top of my quote for himself). And because the clients are already spending money on getting their branding and website done, I can get like $300 for a website, which is realistically like 90 minutes work.
It's been a pretty slow growth, but to be honest if I didn't have this arrangement bringing me in clients, I'd have given up on the search because I had no luck myself. That arrangement has led to me being the writer for a new brand that's starting up soon (based in LA, but I'm in Western Australia), and I'll be approaching her soon with a sort of blog subscription package. X number of blogs over 6 months, pay in advance get 15%. That sort of thing.
So it should be pretty obvious by now that I don't pitch and write interesting articles for magazines or web publications. But I like this set up. Its easy and it can pay really well. But you won't get that sort of arrangement going through one of those sites where you bid for projects.