Quoting Mayo:
"There's one caveat to the aspirin rule, however. Children and teenagers who have certain chronic diseases, such as Kawasaki disease, may need long-term treatment with drugs that contain aspirin. If your child needs aspirin therapy, make sure his or her vaccines are current — including two doses of the varicella (chickenpox) vaccine and a yearly flu vaccine. Avoiding these two viral illnesses can help prevent Reye's syndrome."
And:
"Some hospitals and medical facilities conduct newborn screenings for fatty acid oxidation disorders to determine which children are at greater risk of developing Reye's syndrome. Children with known fatty acid oxidation disorders should not take aspirin or aspirin-containing products."