I am doing research that compares Elite Boarding Schools in the US to
public high schools. My research is focused on the role played by
faculty and staff at the school. My theory is that the organizational
structure (i.e organizational chart, eg. Principal -> Deputy Principal
-> Admissions Director, Education Counselor, Finance Director ->
Teachers, Sports Director, etc) plays a big role in the successful
running of the school, a well defined structure ensures each component
of the school has someone in charge of it. (Unlike public schools
where for example for lack of resources a guidance counselor doubles
up as admissions, and basketball coach) Because Elite schools have
more resources they have the flexibility of developing superb
organizational structures that ensure the school is run successfully.

I am having a hard time finding leads that show the type of
organizational structures at Elite Boarding Schools in the US (i.e
what do they look like). Where can I find sources that will help me
find these structures.

I think you will find that private schools have a lower "over head" of
administrative staff than public schools.  I went to a fine boarding
school in Mass. and they were and still are "light" on administrative
staff vs  public school.  They concentrated on students and teaching
not on layers of administrators.  I think if you look at how many
administrators are in parochial schools systems (eg NY or Chicago
Catholic Schools vs the local Public schools) you will find they have
about an order of magnitude less administrators per 1,000 students
than a public school.

I think their effectiveness is their focus on the customer (the
student and the student's parents) vs anything else.  (The boarding
school I went to was not a Catholic school; I am speaking about
private schools in general, not Catholic schools in particular.)

Each school in run as its own entity and therefore you may have a
difficult time finding out about the administrative stuctures in these
disparate schools.