Can you give a child too many vitamins so that become sick?

YES. Yes, there is a limit to everything and large doses of vitamins can be dangerous and even deadly. Stick to recommended daily doses.
OTC vits usually ok. Generally available vitamines wash out in the urine every day if they are not used by the body so even some excess will be ok. Vitamins a, e, d&k all can store up in the body and excesses can cause a variety of medical problems. It is good to check with your physician any time you use herbals or vitamins. If they are strong enough to do god they are stron enough to do bad in some settings.