The rapid development of effective vaccines against COVID-19 is an extraordinary achievement. However, no medical product can ever be considered risk-free. Several countries have a pharmacovigilance system that detects, assesses, understands, and prevents possible adverse effects of a drug. To benefit from such huge data sources, specialists and researchers need advanced big data analysis tools able to extract value and find valuable insights. This paper defines a general framework for a pharmaceutical data analysis application that provides a predefined (but extensible) set of functions for each data processing step (i.e., data collection, filtering, enriching, analysis, and visualization). As a case study, we present here an analysis of the potential side effects observed following the administration of the COVID-19 vaccines. The experimental evaluation shows that: (i) most adverse events can be classified as non-serious and concern muscle/joint pain, chills and nausea, headache, and fatigue; (ii) the notification rate is higher in the age group 20–39 years and decreases in older age groups and in very young people.