In recent months, many governments have announced COVID-19 vaccination programs and plans to help end the crises the world has been facing since the emergence of the coronavirus pandemic. In Saudi Arabia, the Ministry of Health called for citizens and residents to take up the vaccine as an essential step to return life to normal. However, the take-up calls were made in the face of profound disagreements on social media platforms and online networks about the value and efficacy of the vaccines. Thus, this study seeks to explore the responses of Saudi citizens to the COVID-19 vaccines and their sentiments about being vaccinated using opinion mining methods to analyze data extracted from Twitter, the most widely used social media network in Saudi Arabia. A corpus of 37,467 tweets was built. Vector space classification (VSC) methods were used to group and categorize the selected tweets based on their linguistic content, classifying the attitudes and responses of the users into three defined categories: positive, negative, and neutral. The lexical semantic properties of the posts show a prevalence of negative responses. This indicates that health departments need to ensure citizens are equipped with accurate, evidence-based information and key facts about the COVID-19 vaccines to help them make appropriate decisions when it comes to being vaccinated. Although the study is limited to the analysis of attitudes of people to the COVID-19 vaccines in Saudi Arabia, it has clear implications for the application of opinion mining using computational linguistic methods in Arabic.