Several countries have targeted adolescents for immunization against SARS-CoV-2 to mitigate COVID-19 spread. In Italy, immunization for children ≥ 12 years has been available starting from June 2021. We conducted a cross-sectional study to investigate the knowledge, attitude and intention to vaccinate children < 18 years in Italian families. We used a multinomial logistic regression model to investigate factors associated with intention to vaccinate. We collected a total of 1696 responses. Among the 491 families of children ≥ 12 years, 41.2% would not vaccinate their children and 21.2% were uncertain, while among the 1205 families of children < 12 years, 36.1% would not vaccinate and 33.8% were uncertain. Determinants of intention to vaccinate both age groups were perceived safety and efficacy of vaccines and perceived risk of transmitting infection to adults. For children < 12 years, additional determinants were perceived risk of being infected and being hospitalized because of COVID-19. In view of the expanding strategy to vaccinate adolescents and the availability of immunization for children < 12 years, our results call for a communication strategy targeted at families of children focused on the safety and efficacy of COVID-19 vaccine in children and on the dynamics of infection spread across different age groups. As perceptions in families are volatile and may change rapidly over time, repeated surveys for measuring attitudes to vaccinate would be advisable.