Online public health discourse is becoming more and more important in shaping public health dynamics. Large Language Models (LLMs) offer a scalable solution for analysing the vast amounts of unstructured text found on online platforms. Here, we explore the effectiveness of Large Language Models (LLMs), including GPT models and open-source alternatives, for extracting public stances towards vaccination from social media posts. Using an expert-annotated dataset of social media posts related to vaccination, we applied various LLMs and a rule-based sentiment analysis tool to classify the stance towards vaccination. We assessed the accuracy of these methods through comparisons with expert annotations and annotations obtained through crowdsourcing. Our results demonstrate that few-shot prompting of best-in-class LLMs are the best performing methods, and that all alternatives have significant risks of substantial misclassification. The study highlights the potential of LLMs as a scalable tool for public health professionals to quickly gauge public opinion on health policies and interventions, offering an efficient alternative to traditional data analysis methods. With the continuous advancement in LLM development, the integration of these models into public health surveillance systems could substantially improve our ability to monitor and respond to changing public health attitudes.