What do I do if my child is 16 and has still not gone through puberty?

Call Doc ASAP. No matter what, this is delayed and needs immediate assessment. For either sex, for any reason, 16 is delayed for puberty to have not begun. Call your doc asap...
Schedule a checkup. By 16yrs both genders should have pubertal changes, some of which may not be obvious to parents due to the secretive nature of some adolescents. On general exam your physician can look for signs of emerging pubertal change or begin the process of finding a reason for the delay. Some enter puberty later than their friends, but continue to grow taller thru the many years the full process may take.