According to recent Nutritional Guidelines for Vegetarian Children article in LA Times, it appears that even vegetarian children get protein in their diets, but they can easily miss the very important Vitamin B12, Iron, and Omega 3 fatty acids.
Vitamin B-12: The most easily missed vitamin in a vegetarian diet. It is essential for making DNA and maintaining healthy nerve and red blood cells. It is easily found in a non-vegetarian diet but vegetarians and vegans can get it from fortified cereals or supplements. Talk to your Doctor about getting supplements.
Iron:While Iron is available in beans, tofu, spinach and raisins, it is very hard to absorb it from these foods compared to absorbing it from meat. Iron is very important for children as it allows for greater neural connections and encourages myelin production. Lack of Iron or even subtle shortage of iron can adversely affect intelligence and behavior in children.(Source: Raise a Smarter Child by Kindergarten by David Perlmutter)
So for vegetarian diets one of the best ways to increase absorption is to consume a Vitamin C source the same time you are consuming the plant iron. For instance drink orange juice with your spinach dish.
Omega-3 fatty acids: Very important for brain development of babies and young children. Vegetarians need to eat flax seed, walnuts, avocado as well as eggs and yogurt fortified with Omega-3.