Search career websites and newspapers
You can find career websites that specifically show jobs for non-profit organizations. Visit these websites and apply for any jobs that come your way. Also, check newspapers for any new openings. Most of the non profit organizations requires constant recruitrments, and to ensure a fair system, they regularly advertise the openings in the newspapers.
Get involved in the Non-profit community
You should keep up with any work that non-profit groups are doing in your area or nearby. Keep yourself updated and look for any available opportunities to volunteer. This will immensely help your cause in establishing contacts, thus opening up new avenues for you to find a job.
Search for all such organizations in your vicinity
An effective way to find a job at a non profit organisation is to list down all of them within your city, and then perosnally drop a CV at each one. You might not get a call immidiately, but once there are openings, your CV will facilitate their search.