Aren't there certain health problems people who strictly eat veggies and nothing else run into if they aren't careful?
Other than vitamin B-12 deficiency, it's possible to have a balanced diet and be vegan. That isn't to say that vegans actually all do this. There are vegans with absolutely atrocious diets.
The B-12 thing is because there is basically no good vegan source for this vitamin. Apparently, you store it up and so new vegans can survive off the built up stores of it meat eaters get if they don't have other problems like excessive drinking that deplete B vitamins. So unless you make an exception for this and take some kind of supplement, you're going to end up deficient in this nutrient. It doesn't kill you, but it causes fatigue, headaches, and a number of ill effects.
Other than that, though, pretty much anything else is available from some non-animal source.
What you might have trouble doing is bulking up if you're into bodybuilding, though, because most of the dense protein sources are animal, or like nuts, contain a lot of fat. And again, if you make an exception for things like eggs, egg white protein will probably do you fine.
Not an expert in the subject, but have read a lot of sperging on both sides and trolled arguments by posing as either a fanatical vegan or vegan-hater.