There are many reasons why a child can be breaking out but having the wrong soap or shampoo is a leading cause of allergic reactions. Some have wheat, coconut, and even nuts! Here are links to some recommended products.
Free & Clear makes a great shampoo and body wash that is allergy friendly.
Cetaphil is another great company but be careful which product you buy from them. Cetaphil baby wash contains wheat and their facial cleanser has some added ingredients that can cause some reactions. The most gentle one that can be used for body and shampoo is Cetaphil Gentle Skin Cleanser.
Cerave Baby Wash And Shampoo is another great option.