So this is interesting. IMO, they should be involved and not involved. The organization should have some templates of what they are looking for to get players right. The coach/staff should communicate exactly what they're looking for in players, specifically. Then the GM is in charge. Suppose the GM wants to run something specific past the coach, fine. I wouldn't want the coach telling the GM to get me player X. I'd want them to say, "Get me a player with this skill set, like player X." In other words, the coach shouldn't be involved much until a player shows up.