Of course it's possible.
I'm not vegan, as I occasionally eat fish, but I consume no red meat poultry or dairy, and I only have fish once every couple of weeks, so my diet is close to a vegan diet. I've been raising my daughter the same way since birth. She's now 15 months, and she's doing great.
She was exclusively breastfed for 6 months, and then I started her on whole grain cereal (brown rice, millet, oatmeal) mixed in with expressed breastmilk.
I'll just tell you what she eats, so you can get an idea.
Breakfast: oatmeal(B vitamins, iron) mixed with breastmilk, tahini(lots of calcium), and flax seeds(EFA's).
Lunch: carrot juice, a small piece of avocado and either beans, lentils, chickpeas, tofu, or occasionally fish. All with lots of veggies(celery carrots, zucchini, spinach...) From this meal she's getting lots of calcium, iron, protein, and lots of vitamins.
Evening: vegetables (sweet potatoes, squash, carrots, celery, zucchini, spinach, broccoli, cauliflower, potatoes, beets, etc) with brown rice, or millet, or pasta, or whole grain bread.
Snack: Fruit, or rice cakes, or vegan cookies.
I also still nurse her on demand, so she's getting lots of breastmilk. You can nurse for as long as baby wants. And as long as you nurse, you know he's getting lots of nutrition.
I think you can raise a perfectly healthy child on a vegan diet, provided you make lots of vegetable soups with lots of veggies, veggie and fruit juices, soy shakes with flax seeds, tahini, and fruit, nuts and seeds,whole grains, quinoa, millet, buckwheat, amaranth, great foods loaded with nutrition, like avocados, dandelion, kale, carrots...
I find vegans are usually a lot more informed about nutrition, and therefore a lot healthier, with more vitamins.
Just read about nutrition and find out which foods have the most nutrients, and include them always in your meals.
I hope this helps.
EDIT: To people who suggest that a baby should grow up and choose their diet, I would like to ask, how is that possible?
How can a 4 month old choose to be a vegan, or an omnivore? My parents didn't ask me, they just shoved meat down my throat. Yo;u can't ask your kids. Most kids would live on ice-cream and candy if you let them. That's why parents or good parents, do research about what 's really healthy, and they start their kids on a healthy diet from day 1. Why would you give your baby something you believe to be unhealthy, because you want them to be able to choose?
Some cultures eat dogs. Would you offer your baby dog meat so that they can have the choice? Well then why should a vegan offer their children pork or beef? It's the same thing.