Why do non-governmental organizations tend to publish non peer-reviewed "grey" literature?

I can think of a couple of reasons:

1. Peer review is slow. NGOs want their paper to influence policy now, not in one or two years from now.
2. Peer review journals have a different audience than grey literature. NGOs typically want to communicate with people like policy makers or journalists, and these are less likely to read peer reviewed journals.
3. Related to point 2: in order to effectively communicate with their intended audience they need a different style of writing and presenting their arguments than what is common and acceptable in peer reviewed journals.

Because they are not incentivized to.

Publishing in peer reviewed journals costs time, money, and is frustrating. These costs are more relevant in social sciences with a lower turnaround time than natural sciences, but are nonetheless a factor.

The main reason academics publish in these is because they are incentivized to do so by tenure and status. These incentives come from the need of measuring academics' performance (and that of their institutions). Among other things, academic institutions compete for the best researchers, phd students and funding -- and signal quality with their publications.

NGOs mostly do not have research as an intrinsic motive. The aforementioned competition is much less relevant for them. Therefore, many of them do not incentivize their researchers to publish in peer-reviewed journals. Given the lack of incentives, and the presence of high cost of doing so, few researchers in NGOs publish in peer-reviwed journals.

Expanding on the answer by @Maarten Buis:

I think it is fundamentally about why people in different settings do research.

NGO's usually do not do research for the sake of research. They generally have a social mission to accomplish, thus their research is conducted to that end. It is not to increase knowledge, as it is in academia. To accomplish their mission, they need media, financial and public support. If you need as much as possible of those things, would a fancy infographic work better than a peer-reviewed paper?

Thus, they need to make sure their research is as accessible as possible for their base, and a broad audience helps too. Supporters for NGOs come from every walk of life. Whatever the people around here happen to think, most people are not turned on at the thought of peer-reviewed research, if they even know what it is. The audience of any one peer-reviewed paper is often very small. Peer-reviewed papers are usually locked behind paywalls, use complex terminology and are nearly always so specialised they cannot be accessed, let alone appeal to a broad audience. NGO's probably aren't going to add much to their audience by publishing an extra peer-reviewed paper.

Academics, on the other hand, get pay rises and positions based on peer-reviewed publications, even if the publications happen to be whatever bit of data They could maybe, possibly turn into a peer-reviewed paper. So, cynic that I am, I have to point out that NGOs are not the only people motivated by money.

Edit: As for the added credibility - credibility doesn't have one currency, that being peer-reviewed papers. Something is not made credible by peer-review (especially with the many meta-analyses showing all kinds of problems in the literature, from bad statistics to publishing results selectively leading to positivity bias). NGO's are subject to a whole suite of regulations and audits, not to mention public opinion, that most academics never hear about. So, maybe they don't have peer-reviewed credibility, but they do have other standards.