Building models to detect vaccine attitudes on social media is challenging because of the composite, often intricate aspects involved, and the limited availability of annotated data. Existing approaches have relied heavily on supervised training that requires abundant annotations and pre-defined aspect categories. Instead, with the aim of leveraging the large amount of unannotated data now available on vaccination, we propose a novel semi-supervised approach for vaccine attitude detection, called VADet. A variational autoencoding architecture based on language models is employed to learn from unlabelled data the topical information of the domain. Then, the model is fine-tuned with a few manually annotated examples of user attitudes. We validate the effectiveness of VADet on our annotated data and also on an existing vaccination corpus annotated with opinions on vaccines. Our results show that VADet is able to learn disentangled stance and aspect topics, and outperforms existing aspect-based sentiment analysis models on both stance detection and tweet clustering.