Nutrition during childhood and adolescence is very important for growth and can have long-term health implications. The COVID-19 lockdown caused significant changes in everyday life, including in children and adolescents. This study evaluated the effects of the first COVID-19 lockdown on eating habits and body weight in children and adolescents. An online cross-sectional survey was carried out among the parents of children (5–9 years) and adolescents (10–14 years) living in Italy. The online self-administered questionnaire included demographic and anthropometric data (reported weight and height) weight and dietary habit changes during the COVID-19 lockdown in Italy (March to June 2020). A total of 439 participants were included. We found a change in eating habits with an increase in consumption of sweet packaged snacks (34%) and processed meat (25%), as well as bread, pizza and bakery products (47%). We also found an increase in vegetable, fresh fruit and legume intake (19%), and a reduction in sweet beverage and candy intake. A total of 59.7% of the participants reported body weight gain, with adolescents gaining more than children (67% vs. 55%, p = 0.010, respectively). In children, body weight gain was associated with a change in body height and increased consumption of dairy products and sweet packaged snacks, while in adolescents it was associated with an increased intake of comfort foods and processed meat. Our data highlighted the need to carefully monitor eating behaviors to avoid the establishment of unhealthy eating habits and prevent obesity in children and adolescents during periods of self-isolation.