A very good article regarding vegetarian eating for children and adolescents, from a medical source can be found here. It is quite long and covers the main problems related to possible nutritional deficiencies: protein, Cobalamin (B12), Iron, Zinc, Calcium, Vitamin D, Omega-3 Fatty Acids.
It also covers possible growth and Development Issues.
Short answer is that children can develop normally on a vegetarian diet.
An official statement from an important Dietetic Association can be found here (my emphasis):
It is the position of the American Dietetic Association that appropriately planned vegetarian diets, including total vegetarian or vegan diets, are healthful, nutritionally adequate, and may provide health benefits in the prevention and treatment of certain diseases.Well-planned vegetarian diets are appropriate for individuals during all stages of the life cycle, including pregnancy, lactation, infancy, childhood, and adolescence, and for athletes.