Background: The Oxford–AstraZeneca vaccine (Covishield) was the first to be introduced in Bangladesh to fight the ongoing global COVID-19 pandemic. As this vaccine had shown some side-effects in its clinical trial, we aimed to conduct a study assessing short-term adverse events following immunization (AEFIs) in Bangladesh. Method: A cross-sectional study was conducted on social and electronic media platforms by delivering an online questionnaire among people who had taken at least one dose of the COVID-19 vaccine. The collected data were then analysed to evaluate various parameters related to the AEFIs of the respondents. Results: A total of 626 responses were collected. Of these, 623 were selected based on complete answers and used for the analysis. Most of the respondents were between 30–60 years of age, and 40.4% were female. We found that a total of 8.5% of the total respondents had been infected with the SARS-CoV-2 virus. Our survey revealed that out of 623 volunteers, 317 reported various side-effects after taking the vaccine, which is about 50.88% of the total participants. The majority of participants (37.07%, 231/623) reported swelling and pain at the injection site and fever (25.84%, 162/623); these were some of the common localized and generalized symptoms after the COVID-19 vaccine administration. Conclusion: The side-effects reported after receiving the Oxford–AstraZeneca vaccine (Covishield) are similar to those reported in clinical trials, demonstrating that the vaccines have a safe therapeutic window. Moreover, further research is needed to determine the efficacy of existing vaccines in preventing SARS-CoV-2 infections or after-infection hospitalization.