Each FBS and FCS conference has a coordinator of officials charged with assembling, directing, assigning, and evaluating a staff of officials. The coordinators are hired by, and report to, the commissioners of each conference. They keep their jobs by putting a high quality product on the field. An individual coach or institution, officially, has no right to 'scratch' anybody, or 'prefer' anybody. However, there have been cases where a coach/institution puts enough pressure on the commissioner to not allow an official or a crew to work their future games, that the commissioner, in turn, puts that pressure on the coordinator to do just that. But those are very unusual, if not rare.
May be the same, or very similar for, D2 and below (just not really sure).
Can't speak for any other state, but with regard to Texas public HS varsity football, it is governed by an organization called the UIL, which, for all intents and purposes, is run by the HS ADs and superintendents. Officials for HS sports in Texas are organized by an independent organization called TASO. TASO recruits and trains officials for all major HS team sports, and acts on behalf of member officials in dealing with the UIL. The UIL policy is that officials for any UIL varsity contest are to be mutually acceptable to both institutions. Generally speaking, TASO members work all UIL contests, assigned by TASO assignors in local areas throughout the state (i.e., chapters). How assignments are made varies widely among the chapters. But, the larger schools tend to have drafts of officials/crews from chapters chosen by the schools, to work their contests. The small schools more often get officials assigned directly by the assignor from the Chapter(s) of their choice.
The UIL's "mutually accceptable" policy means that schools can, effectively, 'scratch' officials or crews. They can also have 'preferred' officials/ crews.
FYI, the UIL compensation system is based upon the paid attendance at each contest. So, very generally, the bigger the school, the bigger the crowd, the bigger the check. So, that is the golden carrot that the schools dangle over officials. (Want our games? Be nice to us.). I hope with all my soul that no TASO official would ever deliberately 'assist' a school, in exchange for continued assignments with any institution.
Some chapters say that they do not allow schools to select or scratch officials. The schools get the officials that the Chapter assigns. If so, good for them, and I hope they are able to maintain that system. But it isn't the UIL policy.
But, that's our dilemma. Until the schools are taken out of the selection process, we will always have the specter of potential for conflicts of interest. But, that would probably take a literal act of the state legislature to fix that. Good luck.
Robert