Does a child develop lactose intolerance or are people just born with it?

Both. Lactose intolerance is thought to have three basic pathways: lack of the enzyme lactoase from birth (the largest group); loss of the lactase enzyme over time (a lesser number) & transient lactose intolerance due to localized bowel injury (can occur in anyone often with bad diarrhea.).
Lactose intolerance. The answer is both yes and no. Lactose intolerance is caused by a deficiency of the enzyme called lactase that breaks down the sugar in milk called lactose. Some persons are born with lactase deficiency and so, are lactose intolerant. One can develop a transient lactose intolerance by loss of the enzyme during bouts of prolonged diarrhea.