The outbreak of the COVID-19 pandemic has also triggered a tsunami of news, instructions, and precautionary measures related to the disease on social media platforms. Despite the considerable support on social media, a large number of fake propaganda and conspiracies are also circulated. People also reacted to COVID-19 vaccination on social media and expressed their opinions, perceptions, and conceptions. The present research work aims to explore the opinion dynamics of the general public about COVID-19 vaccination to help the administration authorities to devise policies to increase vaccination acceptance. For this purpose, a framework is proposed to perform sentiment analysis of COVID-19 vaccination-related tweets. The influence of term frequency-inverse document frequency, bag of words (BoW), Word2Vec, and combination of TF-IDF and BoW are explored with classifiers including random forest, gradient boosting machine, extra tree classifier (ETC), logistic regression, Naïve Bayes, stochastic gradient descent, multilayer perceptron, convolutional neural network (CNN), bidirectional encoder representations from transformers (BERT), long short-term memory (LSTM), and recurrent neural network (RNN). Results reveal that ETC outperforms using BoW with a 92% of accuracy and is the most suitable approach for sentiment analysis of COVID-19-related tweets. Opinion dynamics show that sentiments in favor of vaccination have increased over time.