Yes Kawasaki disease does run in families.
If the parent or parents have had Kawasaki disease as a child then the child of those parents or parent is likely to also get Kawasaki disease.
It doesn't always mean the child of a parent who had Kawasaki disease will get the Kawasaki disease themselves but there is a higher chance of a child becoming infected with the Kawasaki disease if one or both of their parents had the Kawasaki disease as a child.
The Kawasaki disease may also be inherited through other family members as well.
If someone in your family has had Kawasaki disease as a child then the child is twice as likely to get Kawasaki disease than a child who has no Kawasaki disease in the family.