Now-a-days, social media platforms enable people to continuously express their opinions and thoughts about different topics. Monitoring and analyzing the sentiments of people is essential for governments and business organizations to better understand people's feelings and thoughts. The Coronavirus disease 2019 (COVID-19) has been one of the most trending topics on social media over the last two years. Consequently, one of the preventative measures to control and prevent the spread of the virus was vaccination. A dataset was formed by collecting tweets from Twitter for over a month from November 13th to December 31st, 2021. After data cleaning, the tweets were assigned a positive, negative, or neutral label using a natural language processing (NLP) sentiment analysis tool. This study aims to analyze people's public opinion towards the vaccination process against COVID-19. To fulfil this goal, an ensemble model based on deep learning (LSTM-2BiGRU) is proposed that combines long short-term memory (LSTM) and bidirectional gated recurrent unit (BiGRU). The performance of the proposed model is compared to five traditional machine learning models, two deep learning models in addition to state-ofthe-art models. By comparing the results of the models used in this study, the results reveal that the proposed model outperforms all the machine and deep learning models employed in this work with a 92.46% accuracy score. This study also shows that the number of tweets that involve neutral, positive, and negative sentiments is 517496 (37%) tweets, 484258 (34%) tweets, and 409570 (29%) tweets, respectively. The findings indicate that the number of people carrying neutral sentiments towards COVID-19 immunization through vaccines is the highest among others.