Can children grow normally on a vegetarian diet?
Is it safe? Or are there any known risks?
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.
Yes, it is safe to feed your children the vegetarian diet.
I have grown up as a vegetarian. Most of the meals would include
- Stir-fried vegetables
- Freshly made Chapati/Indian bread (from the wheat/chickpea/Bajra, Jowar and Soya ground into flour at home)
- Enough pulses or lentils (Yellow pigeon lentil, Green gram, red lentils to name a few)
- Beans (Mung beans, kidney beans, Chickpeas, Black-eyed peas, Dew gram beans)
- For sweets, we never had anything made up of food-grade gelatin (whose preparation requires animal fat). It was mostly Dates with Ghee/ clarified butter, sweet buckwheat halwa with jaggery and clarified butter.
- Nuts like overnight water-soaked peanuts, cashews, and almonds were amongst the first things that we ate in the morning.
- Seeds like Linseeds and roasted coriander seeds were eaten after every meal
- Milk, Yogurt and Cheese were also included in daily meals
- Spices like Kurkuma, Coriander, and Cumin powder, Black pepper, Rock salt, black salt, Cinnamon, Cardamom, Nutmeg were also added to most of my daily meals.
Let me include one more thing. Please please test your children for all possible deficiencies and accordingly add supplements to their diet. (Remember that supplements cannot replace the diet itself) I have grown to be a healthy person and so are many people living around me. I think it is completely safe to feed children with a vegetarian diet. :)
The American Dietetic Association and the British Dietetic Association, the largest bodies of nutrition and diet professionals in both countries--tens of thousands of professionals--state that
a plant-based (vegan) diet is nutritionally adequate and safe for all stages of life.
Academy of Nutrition and Dietetics
It is the position of the Academy of Nutrition and Dietetics that appropriately planned vegetarian, including vegan, diets are healthful, nutritionally adequate, and may provide health benefits for the prevention and treatment of certain diseases. These diets are appropriate for all stages of the life cycle, including pregnancy, lactation, infancy, childhood, adolescence, older adulthood, and for athletes. 
The United States Department of Agriculture
Vegetarian diets (see context) can meet all the recommendations for nutrients. The key is to consume a variety of foods and the right amount of foods to meet your calorie needs. Follow the food group recommendations for your age, sex, and activity level to get the right amount of food and the variety of foods needed for nutrient adequacy. Nutrients that vegetarians may need to focus on include protein, iron, calcium, zinc, and vitamin B12. 
Dietitians of Canada
A well planned vegan diet can meet all of these needs. It is safe and healthy for pregnant and breastfeeding women, babies, children, teens and seniors. 
The Heart and Stroke Foundation of Canada
Vegetarian diets (see context) can provide all the nutrients you need at any age, as well as some additional health benefits. 
The British National Health Service
With good planning and an understanding of what makes up a healthy, balanced vegan diet, you can get all the nutrients your body needs. 
The British Nutrition Foundation
A well-planned, balanced vegetarian or vegan diet can be nutritionally adequate […] Studies of UK vegetarian and vegan children have revealed that their growth and development are within the normal range. 
The Dietitians Association of Australia
Despite these restrictions, with good planning it is still possible to obtain all the nutrients required for good health on a vegan diet. 
National Health and Medical Research Council
Alternatives to animal foods include nuts, seeds, legumes, beans and tofu. These foods increase dietary variety and can provide a valuable, affordable source of protein and other nutrients found in meats. These foods are also particularly important for those who follow vegetarian or vegan dietary patterns. Those following a vegetarian diet can still meet nutrient requirements if energy needs are met and the appropriate number and variety of serves from the Five Food Groups are eaten throughout the day. For those eating a vegan diet, supplementation of B12 is recommended. 
A well-planned vegetarian diet can meet the needs of people of all ages. The key is to be aware of your nutritional needs so that you plan a diet that meets them. 
Harvard Medical School
Traditionally, research into vegetarianism focused mainly on potential nutritional deficiencies, but in recent years, the pendulum has swung the other way, and studies are confirming the health benefits of meat-free eating. Nowadays, plant-based eating is recognized as not only nutritionally sufficient but also as a way to reduce the risk for many chronic illnesses. 
Sources are listed at CalculatedVegan.wordpress.com/authority-argot
Lastly, there is a difference between dietary risks and dietary hazards: Wageningen University - Difference between hazard and risk (video)
I can give you the example of two of my friends who have raised their daughters on a vegetarian diet. One of the daughters is 15 now, nicely developed, tall and bright and the other one is 10 years and also very normal and nicely developed.
My experience has been that vegetarian parents tend to watch what their children eat way more closely. They know that they are the odd ones out and are focused on possible nutrient deficiencies, whereas 'regular' diet parents (with meat in the diet) just go along as ever, with not so much extra focus on deficiencies. The obvious risk is vitamin B12 deficiency., but if you get your children tested at a doctor's office once a year, you can counteract that very easily.
Most vegetarians I know also watch closely over the quality of food. They would only give their family organic food for example, which alone is a good thing. But of course you can also live on a healthy diet with organic meat, sure. But this was not your question. In my experience with my friends, it is totally okay to feed your kids a vegetarian diet.
I agree, a vegan diet is adequate also for children, I am nutritionist and some of my clients are vegan children, they flourish better than others, but one thing ist extremely important: a vegan diet alway has to be balanced. It is very important to know what to eat and how much of it, and for children this is even more important than for adults. So please go to a good vegan nutritionist.