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SARS-CoV-2 and its mutations are rapidly spreading around the world threatening human population with millions of infections and deaths. Vaccines are the only available weapon at hand to mitigate the spread. As a result, the development of efficient systems to understand and supervise the information dissemination as well as the sentiment evolution toward vaccines is critical. The goal of this research was to build and apply a supervised machine learning approach to monitor the dynamics of public opinion on COVID-19 vaccines using Twitter data. 1,394,535 and 61,077 tweets in English and Greek about COVID-19 vaccines, respectively, were collected, classified based on sentiment polarity, and analyzed over time to gain insights into sentiment trends. The findings reveal that overall negative, neutral, and positive sentiments were at 36.5%, 39.9% and 23.6% in the English language dataset, respectively, whereas overall negative and non-negative sentiments were at 60.1% and 39.9% in the Greek language dataset. Policy makers and health experts should take into consideration social media sentiment analysis alongside other ways of evaluating public sentiment. Social media users are actively seeking and sharing information about all pandemic-related topics, allowing governments to use social media not only to better inform the public with accurate and reliable news, but also alleviate disease-specific concerns, minimize the distribution of fake news, and develop effective crisis management strategies.