Objectives: This study aims to assess COVID-19 vaccine acceptance, uptake, and hesitancy among parents and caregivers of children in Saudi Arabia during the initial rollout of pediatric COVID-19 vaccination. Methods: An electronic survey was used to collect data from participants who visited a COVID-19 vaccine center. The survey included demographic data, COVID-19 vaccine status among participants and their children, and reasons for vaccine acceptance or rejection. The Vaccine Hesitancy Scale (VHS) tool was also employed to assess vaccine hesitancy and attitudes toward the COVID-19 vaccine and routine childhood vaccination. Multivariate binary regression analysis was used to identify predictors of actual COVID-19 vaccine uptake among children. Results: Of the 873 respondents included in the analysis, 61.5% were parents and 38.5% were other caregivers. Of the participants, 96.9% had received the COVID-19 vaccine. Six hundred and ninety-four participants accepted the vaccine for their children, with the main reasons being an endorsement by the Saudi Ministry of Health (60%) and the importance of going back to school (55%). One hundred and seventy-nine participants would not vaccinate their children, with the most common reasons being fear of adverse effects (49%) and inadequate data about vaccine safety (48%). Factors such as age, COVID-19 vaccination status, self-rated family commitment level, attitudes toward routine children’s vaccines, and participants’ generalized anxiety disorder (GAD7) score did not significantly correlate with children’s COVID-19 vaccination status. Parents were less likely to vaccinate their children compared to other caregivers, and participants with a higher socioeconomic status were more likely to vaccinate their children. Conclusion: Vaccine acceptance and uptake were high during the initial pediatric COVID-19 vaccination rollout in Saudi Arabia. Still, the ongoing endorsement of the Ministry of Health and healthcare authorities should continue to advocate for better vaccine uptake in children.