SARS-CoV-2 and its mutations are spreading around the world, threatening the human population with millions of infections and deaths. Vaccines are considered the main 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 evolution of sentiments towards vaccines is critical. The goal of this research was to build and apply a supervised learning approach to monitor the dynamics of public opinion on COVID-19 vaccines using Twitter data. 1,394,535 and 61,077 tweets about COVID-19 vaccines, respectively in English and Greek, were collected, classified based on sentiment polarity and analyzed over time to gain insights into sentiment trends. Our 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. Policymakers and health experts could take into consideration social media sentiment analysis alongside other ways of evaluating public sentiment. Social media users are actively seeking and sharing information about pandemic-related topics, allowing governments to use social media to develop effective crisis management strategies, better inform the public with accurate and reliable news, and alleviate disease-specific concerns.