How do you go about trying to track the conversion rate on organic advertising-cum-Reddit post? It seems like it'd be impossible without the consent of Reddit.

You would use different metrics. It costs nothing to make the post so you can just pump them out, you don't need to use impressions as a metric anymore because you're not paying for impressions. Instead you get good at curating titles, images and articles that gain traction on the site. This isn't a wild idea, it's all over the reddit frontpage every day.