Working as data scientist for a nonprofit company
I see at least five ways to approach this problem of finding a data scientist position/work specifically at non-profit, non-governmental or similar organizations, as I describe below. I hope that this is helpful.


First, and the most obvious, way is to search major job portals, such as indeed.com, dice.com, monster.com, CareerBuilder, Glassdoor and others, for data scientist or similar positions, such as data analyst, data engineer, quantitative analyst, statistical modeler, or even market researcher.
The second and also obvious way is to tap your professional social networks and research or inquire your contacts for any potential data science work opportunities in the areas of your interests.
The third way is to search web sites, focused on the non-profit and related topics (they usually have a job listings or similar section) as well as specialized non-profit job portals. For example: http://encore.org, http://idealist.org, http://bridgespan.org, http://cgcareers.org, http://opportunityknocks.org, http://foundationcenter.org, http://thenonprofittimes.com, http://philanthropy.com, http://ynpn.org, http://philanthropyjournal.org, http://nonprofit-jobs.org, CoF Jobs (section isn't easy to find), http://careers.councilofnonprofits.org, http://nonprofittalentmatch.com.
The fourth way is to perform Internet search and research on non-profit or similar organizations that you might be interested in working with (criteria might vary from organization's size to industry focus or geographical locations). Based on the information presented  on their websites, make notes and then approach those organizations, directly inquiring or applying for positions of interest.
The fifth way is to consider various non-profit, for-profit and otherwise social good-themed data science-focused organizations, initiatives and Kaggle-like competitions, such as DataKind, DrivenData, DataLook and Data Science for Social Good.