If I give my children a pretty balanced diet, are there any particular vitamins they won't get enough of?

D... ...Tends to be the vitamin most often deficient in a normal diet. It's just too difficult to eat enough fish or drink enough milk for adequate levels of d to be maintained.
Probably not. Well balanced diets generally give your children the essential vitamins and minerals they need to grow at a healthy rate. However, vegan and vegetarian diets may be missing some of the essential vitamins (such as B12 in a vegan diet). If your child is very picky and doesn't eat a well balanced diet a multivitamin is a good idea.