Objective: The exponential growth of digital social platforms has not only connected individuals globally but has also provided a platform for users to freely express their experiences and viewpoints on topics spanning from consumer products and services to broader societal matters such as political issues. Within this expansive digital discourse, in the recent years, one notably discussed subject has been the SARS-CoV-2 (COVID-19) vaccines. In this article, our focus is on investigating the profound impact of neural networks in the analysis of sentiments expressed by people concerning the introduction and utilization of these vaccines. Background: Sentiment analysis, a critical facet of natural language processing (NLP), is replete with intricate associations in the linguistic landscape. Within its realms, many sophisticated methodologies, such as machine learning algorithms, including neural network architectures, are employed to decipher the intricate web of semantic relationships embedded in textual data, which include, but are not limited to, social media posts. From gathering business intelligence, to market research and competitor analysis, examining sentiments has found many practical uses. In the domain of COVID-19 vaccines, sentiment analysis has provided valuable insights into vaccine hesitancy, vaccine adoption rates, and public trust in the governmental setup and in the pharmaceutical industry. Methods: A systematic literature review (meta-analysis) was carried out to quarry scientific research on neural network-based analysis of sentiments about COVID-19 vaccines. Implementing a thorough search strategy, we isolated relevant articles and methodically examined them to discern key insights that contributed to our comprehension of the utility of neural networks in analyzing public opinion regarding COVID-19 vaccines. Conclusion: Our study provides insights affirming that neural networks have shown a surpassing capacity to discern intricate patterns within vast textual datasets. Their inherent ability to capture contextual nuances in language has enabled a nuanced understanding of diverse sentiments about COVID-19 vaccines. This has helped formulate strategies to alleviate negative sentiments about the vaccines leading to higher vaccine acceptance rates and management of the pandemic.