Monitoring vaccine behaviour through social media can guide health policy. We present a new dataset of 9471 tweets posted in Australia from 2020 to 2022, annotated with sentiment toward vaccines and also 5C, the five types of behaviour toward vaccines, a scheme commonly used in health psychology literature. We benchmark our dataset using BERT and Gradient Boosting Machine and show that jointly training both sentiment and 5C tasks (F1=48) outperforms individual training (F1=39) in this highly imbalanced data. Our sentiment analysis indicates close correlation between the sentiments and prominent events during the pandemic. We hope that our dataset and benchmark models will inform further work in online monitoring of vaccine behaviour. The dataset and benchmark methods are accessible online.