In 2019, a novel severe acute respiratory syndrome (SARS-CoV-2 (COVID-19)) caused a global pandemic. There was an urgent need to develop a vaccine against COVID-19 to reduce its spread and economic burden. The main objective of this study was to understand the attitudes and concerns of healthcare workers (HCWs) towards the upcoming COVID-19 vaccine, whether their decision was influenced by their history of taking the seasonal influenza vaccine, and factors that influence the acceptance of the upcoming COVID-19 vaccine. This was a cross-sectional study conducted in Riyadh, Saudi Arabia. We selected and surveyed 356 HCWs via an electronic self-administered questionnaire. A total of 61.16% of HCWs were willing to receive the COVID-19 vaccine, and 55.9% of them had received the seasonal influenza vaccine in the preceding year (2019–2020). The strongest predictors for taking the COVID-19 vaccine were the HCWs’ belief that the COVID-19 vaccine would be safe, needed even for healthy people, that all HCWs should be vaccinated against COVID-19, and that HCWs will have time to take the vaccine. Being female, being middle aged, having <5 years of work experience, having no fear of injections, and being a non-smoker were predictive factors for taking the upcoming COVID-19 vaccine. No associations were found between the intention to take the COVID-19 vaccine and a history of taking the seasonal influenza vaccine.