Can a child with peanut allergies eat a bar of white chocolate?

Usually. White chocolate does not contain any peanut or nut components.
If not procesed w/pn. White chocolate is made basically the same way as regular chocolate but it lacks the cocoa solids in it. You need to look at the wrapper if it is processed in a plant that deals with peanuts, it is best to stay away.
Read labels. It is prudent to check the label to determine if the candy was made in a facility that uses peanuts. Cross contamination is a possibility if peanuts were used in the facility.