Automated methods for analyzing public opinion have grown in popularity with the proliferation of social media. While supervised methods can be very good at classifying text, the dynamic nature of social media discourse results in a moving target for supervised learning. Meanwhile, traditional unsupervised techniques for extracting themes from textual repositories, such as topic models, can result in incorrect outputs that are unusable to domain experts. For this reason, a non-trivial amount of research on social media discourse still relies on manual coding techniques. In this paper, we present an interactive, humans-in-the-loop framework that strikes a balance between unsupervised techniques and manual coding for extracting latent arguments from social media discussions. We use the COVID-19 vaccination debate as a case study, and show that our methodology can be used to obtain a more accurate, interpretable set of arguments when compared to traditional topic models. We do this at a relatively low manual cost, as 3 experts take approximately 2 hours to code close to 100k tweets.