My son to this day is what I call a 'grazer'. He is not big into breakfast, but will also get hungry at 10- 10:30am.
I give him two choices of a breakfast, but I do notice that after just waking up, he is not really hungry and it takes nearly 35 minutes for a bowl of cereal or a piece of toast.
However, I do not eat breakfast with him or very often. This may be why he does this. He is modeling himself after what he sees in the home. Do you eat breakfast in the morning? (Not many folks do...and I realize the importance of it and when I am really hungry - I will eat.).
Kids will eat when they are hungry and the best practice to have IS to offer breakfast, snack, lunch, snack and then dinner (and possibly a last snack before bed if you eat rather early). This is the best for any body's metabolism. And when I say 'snack' I am referring to healthy choices.
I make sure he has water available at all times as when the body is 'hungry', that is typically a signal that the body needs water as well.
But he does eat... if it's not first thing upon rising, he's usually hungry within the hour and a half. If he doesn't eat much at breakfast, the snack is smaller and of my choosing as he wants something I normally would not give him (sweets).
As for school, it's still a good idea to provide and encourage breakfast time (and eat with her). Even in Kindergarten, they still offer snack. She'll change habits by the time this is no longer allowed (by first grade).