(1) Background: COVID-19 vaccination campaigns offer the best hope of controlling the pandemic. However, the fast production of COVID-19 vaccines has caused concern among the general public regarding their safety and efficacy. In particular, patients with chronic illnesses, such as celiac disease (CD), may be more fearful. Information on vaccine hesitancy plays a pivotal role in the development of an efficient vaccination campaign. In our study, we aimed to evaluate COVID-19 vaccine hesitancy among Italian CD patients. (2) Methods: an anonymous questionnaire was sent to CD patients followed at our tertiary referral center for CD in Milan, Italy. Patients were defined as willing, hesitant and refusing. We evaluated the reasons for hesitancy/refusal and the possible determinants, calculating crude and adjusted odds ratios [AdjORs] with 95% confidence intervals [CIs]. (3) Results: the questionnaire was sent to 346 patients with a response rate of 29.8%. Twenty-six (25.2%) of the 103 respondents were hesitant, with a total refusal rate of 4.8%. The main reason was fear of adverse events related to vaccination (68.2%). Among hesitant patients, 23% declared that their opinion was influenced by their CD. The determinants positively influencing willingness to be vaccinated against COVID-19 were adherence to a GFD, perception of good knowledge about COVID-19 and its vaccines, and a positive attitude to previous vaccines (AdjOR 12.71, 95% CI 1.82–88.58, AdjOR 6.50, 95% CI 1.44–29.22, AdjOR 0.70, 95% CI 0.11–4.34, respectively). (4) Conclusions: CD patients should be vaccinated against COVID-19 and a specific campaign to address the determinants of hesitancy should be developed.