Background: Large-scale vaccination was crucial to address the global COVID-19 pandemic and its associated health risks, including fatal and disabling diseases. However, there were significant challenges to be overcome to ensure the safe and effective implementation of the vaccination program. The aim of the present study was to assess patient safety threats related to the anti-COVID-19 large-scale vaccination process. Methods: Between February and May 2021, we conducted a proper analysis to proactively identify risks and potential Failure Modes (FMs) in the COVID-19 vaccination process using the Failure Mode, Effects, and Criticality Analysis (FMECA) technique at an Italian Public Health Authority. A standardized risk scoring system was used to assess the severity, frequency, and detectability of events associated with potential failures. Criticalities were identified in both the preparatory and operational areas of the vaccination process, and several potential FMs were listed in descending order of risk score (Risk Priority Number, RPN) to ensure prioritization of interventions. Results: The most critical steps were found to be in the operational area rather than in the preparatory one. The highest RPNs were associated with failure or inadequate management of severe allergic reactions that can lead to serious harm and even death of the vaccinated person (RPN 60) and failure to keep updated vaccination teams’ knowledge (RPN 36). Conclusions: Ensuring patient safety and effective clinical risk management are crucial in mass vaccination campaigns. By prioritizing these aspects through collaboration with various stakeholders and implementing preventive measures, patient trust—on which vaccination campaign success relies—can be built and maintained.