The superintendent. However, who do you think hires/fires the superintendent? The elected board. The key word is elected. That's where the problem will be in doing this.
The superintendent runs the day to day operations of the district, and is the highest level on this. However, the board makes policy, which guides, empowers, and limits the superintendent. I'll be looking at board policy in my district on this. My hunch is that they have not directly addressed firearms, except in the case of students. If they did, they'd have to make explicit allowance for police officers and security. They'd also have to be consistent with the actual law on the matter.
Superintendents who make decisions that get unwanted attention for the board typically don't last long. (Hell, they typically don't last long, anyway.) The supt would be advised to convince them first, before making that decision, assuming board policy allows him or her to make it.