Yes, a 5.56 x 45 mm NATO will penetrate the approximately 2 inches of a heavy wood door:
‘Steel’ exterior doors are typically a thin gauge mild steel over a wooden or fiberglass core. You can actually stick an axe through one with enough force. There are ‘bullet-resistant’ steel doors that have a high density composite but you won’t find these in schools because they are quite expensive. A solid steel door would weight far too much to be easily opened and closed and would prevent easy egress in an emergency.
ETA: Ninja’d on both options. I have to type faster.
Stranger