What are the best charities working in vegan outreach?

Here's my understanding of the current state of evidence, keep in mind that I am not a researcher or grantmaker:

1. To my knowledge there is no scientifically rigorous experiment showing that some intervention has a statistically significant effect on the number of vegans.
2. Vegan education organisations also don't tend to report the number of counterfactual vegans they create, to some extent because of measurement difficulties.
3. My guess is that most effective ways(having conversations about veganism with people who trust you) of spreading veganism can't be funded to scale up.
4. Probably education initiatives produce small effects but we don't have sufficiently powered studies to catch these effects. So we have very little data to compare vegan education initiatives to each other.

I don't really have a concrete answer for you, however I am also interested in this topic and have some resources which may be of use to you, namely this article by Social Change Lab on how philanthropists can support social movements, and this forum post, along with the other posts of the series, on how EA can incorporate an "abolitionist" frame of animal advocacy alongside the current advocacy strategies that are more canonical in the EA movement currently (more quantifiable/measurable and "welfarist"), and this post of that series provides some potential avenues for animal advocacy from a more "abolitionist" outlook which may potentially prove to be effective. On top of this there is another forum post about promoting Monitoring, Evaluation and Learning within the broader animal advocacy movement where it generally lacks relative to the more "EA-aligned" animal advocacy.

Maybe this is a bit of an aside (though still probably relevant), but I have been involved in the Plant-Based Universities campaign at my university over the past year (so I may be a little biased here) and this is one initiative which is not exactly aiming to change peoples' attitudes towards animals and veganism directly, but which aims to make change on the institutional level, yet a large part of this does still involves individual outreach among students. This campaign, although it seeks transformative (as opposed to incremental) change, is still in some sense quantifiable as you can see how many universities have transitioned (though of course the broader aim is to bring about a societal shift in attitudes, which is more difficult to quantify yet undoubtedly holds much positive value). There exists a similar initiative in the US in Allied Scholars for Animal Protection (though they also focus on many other initiatives too).

Hopefully this is of value to you, even if I kind of danced around an actual answer to your question.

I'd note that vegan outreach, as defined in your post,^[1] is only one type of demand-reduction strategy that could be employed. Other options might include direct advocacy with the goal of reducing the listener's meat consumption, advocacy for meat taxes to force consumers to partially internalize welfare and environmental costs (driving down demand), and so on. 

The "[m]any vegans" to whom your first paragraph refers may be disposed to vegan outreach over other forms of demand reduction for ideological reasons rather than because they view them as more effective per dollar spent. (Or maybe they do believe other demand-reduction strategies are less effective.) Based on what we know about behavior change more generally, I would guess that getting four people to reduce meat consumption by 25% would be easier than persuading one to reduce it by 100% -- but I could easily be wrong.

That being said, my starting point would be that your money would go further in middle-income countries, both because work there is cheaper and because less of the population potentially open to veganism may have been exposed to the message yet.

1. ^{^}

   I.e., "direct advocacy to individual members of the public with the aim of persuading them to become vegan."