Will exercise help a child grow up to be taller?

No. But exercise is, nonetheless, an extremely important part of a child's life and health. Height is determined by genetics and nutrition. Exercise can help a child be more healthy, more active, have more endurance, stand up straight, and feel well overall - all of those things are very important!
Height is determined by genetic factors, good health and adequate nutrition. Teaching your child healthy habits such as regular exercise and good nutrition will help your child reach his or her maximum genetic potential. Exercise alone, cannot make your child become taller than his or her potential. Not surprisingly, children usually grow in manners similar to their parents.