This thread is a few weeks old but I wanted to reply anyway, hope that's alright.
I have heard of vegan children. Many, if not most, parents who are vegan themselves are going to want to raise vegan children. I can't imagine vegan parents serving hamburger for dinner to their kids while they eat vegan. Vegetarian children are probably more common because vegetarianism itself is more common, for a variety of reasons (relative ease compared to veganism, cost, morality boundaries, etc.)
My daughter, when she was between the ages of 2-5, had a myriad of allergies. Milk, pork, eggs, beef, fish, shellfish, maple, plus about 10 others...thankfully no gluten or soy because then she would've just eaten lettuce probably for those years lol. She ate a lot of tofu and other vegetarian options. Now she's outgrown those allergies, thankfully. It's scary having all of those.
Also, I don't think it's a matter of "training" kids to be vegetarian, as much as it's probably about showing them and telling them why the lifestyle is a good match for their family and what it means to be a vegetarian.