Do people want to be vaccinated against COVID-19? Herd immunity is dependent on individuals’ willingness to be vaccinated since vaccination is not mandatory. Our main goal was to investigate people’s intention to be vaccinated and their intentions to vaccinate their children. Moreover, we were interested in understanding the role of the personal characteristics, psychological factors, and the lockdown context on that decision. Therefore, we conducted an online survey during the lockdown in Portugal (15 January 2021 until 14 March 2021). Participants completed a socio-demographic questionnaire, questions about their intentions of being vaccinated, concerns about the vaccine, a COVID-19 attitudes and beliefs scale, a COVID-19 vaccine attitudes and beliefs scale, and the Domain-Specific Risk-Taking (DOSPERT) Scale. Our results showed that from the 649 participants, 63% of the participants reported being very likely to have the vaccine, while 60% reported being very likely to vaccinate their children. We conducted two linear regression models, explaining 65% of the variance for personal vaccination and 56% of the variance for children vaccination. We found that the COVID-19 vaccine general beliefs and attitudes were the main determinants of vaccination intention. Additionally, our proposed artificial neural network model was able to predict with 85% accuracy vaccination intention. Thus, our results suggest that psychological factors are an essential determinant of vaccination intention. Thus, public policy decision makers may use these insights for predicting vaccine hesitancy and designing effective vaccination communication strategies.