Background: Coronavirus disease 2019 (COVID-19) vaccination hesitancy is a threat as COVID-19 vaccines have reduced both viral transmission and virus-associated mortality rates, particularly in high-risk subgroups. Solid organ transplant recipients (SOTRs) are particularly vulnerable, as the underlying causes of their organ failure and the chronic immunosuppression are associated with a lower immune response to COVID-19 vaccines, and with an excessive risk of death due to SARS-CoV-2 infection. We aimed to evaluate COVID-19 vaccination hesitancy and its reasons in a population of SOTRs. Methods: All the SOTRs attending our post-transplant clinics were asked to fill in a vaccination status form with specific validated questions related to their willingness to receive a third vaccine dose. In the case of negative answers, the patients were encouraged to explain the reasons for their refusal. Among the SOTRs (1899), 1019 were investigated (53.7%). Results: Overall, 5.01% (51/1019) of the SOTRs raised concerns regarding the future third dose vaccination. In more detail, hesitancy rates were 3.3% (15/453), 4.2% (7/166), and 7.3% (29/400) among the investigated liver, lung, and kidney transplant recipients, respectively (p = 0.0018). The main reasons for hesitancy were fear of adverse events (30/51, 58.8%) and perceived lack of efficacy (21/51, 41.2%). Conclusions: Full adherence to ongoing or future vaccination campaigns is crucial to prevent, or at least reduce, COVID-19-related morbidity and mortality in fragile patients. The identification of the reasons influencing COVID-19 vaccination hesitancy in these patients is very important to establish appropriate and targeted patient–doctor communication strategies, and to further implement specific vaccination campaigns.