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.