How do you build up a child's immune system?

Play and explore. Just as you can't read without being exposed to words, the immune system needs exposure -- yes, this includes microscopic germs that you can't see -- to learn and build defenses. So, fill their tummies with a healthy diet and let them play with other children and explore their surroundings. Not only will this enrich their minds, it will make their bodies stronger, inside and out. And vaccinate.
Rest and Nutrition. The best way to build a child's immune system is to provide them a balanced nutritious diet, insure that get plenty of rest and sleep, plenty of fluids, exercise and a good environment that is not stress producing. A daily multiple vitamin may help.