Background Advances in machine learning (ML) models have increased the capability of researchers to detect vaccine hesitancy in social media using Natural Language Processing (NLP). A considerable volume of research has identified the persistence of COVID-19 vaccine hesitancy in discourse shared on various social media platforms. Methods Our objective in this study was to conduct a systematic review of research employing sentiment analysis or stance detection to study discourse towards COVID-19 vaccines and vaccination spread on Twitter (officially known as X since 2023). Following registration in the PROSPERO international registry of systematic reviews, we searched papers published from 1 January 2020 to 31 December 2023 that used supervised machine learning to assess COVID-19 vaccine hesitancy through stance detection or sentiment analysis on Twitter. We categorized the studies according to a taxonomy of five dimensions: tweet sample selection approach, self-reported study type, classification typology, annotation codebook definitions, and interpretation of results. We analyzed if studies using stance detection report different hesitancy trends than those using sentiment analysis by examining how COVID-19 vaccine hesitancy is measured, and whether efforts were made to avoid measurement bias. Results Our review found that measurement bias is widely prevalent in studies employing supervised machine learning to analyze sentiment and stance toward COVID-19 vaccines and vaccination. The reporting errors are sufficiently serious that they hinder the generalisability and interpretation of these studies to understanding whether individual opinions communicate reluctance to vaccinate against SARS-CoV-2. Conclusion Improving the reporting of NLP methods is crucial to addressing knowledge gaps in vaccine hesitancy discourse.