You could use "vertical". As in "I've arranged the children in vertical order". I wouldn't call it common phrasing but I believe it would be well understood. In most cases I think common phrasing would be "in order of height".
I'm surprised people are confusing the idea of a "stack" and "vertical". Stack can mean many things some connected with the notion of the vertical, some not. You can stack horizontally or call a random pile a stack. When I see vertical, I think perpendicular to the horizonal, upright, directly overhead, height. "He had me at a vertical advantage", is often a dole way to reference a height difference. The context stresses the difference not the reason for the difference. Perhaps one is a giant, or perhaps a child is "stacked" on the shoulders of a parent. In the SO's question the context was a picture, it seems difficult for me to think people would misunderstand the concept of "vertical" if they saw the children side by side and not "stacked". Granted, I tend to think more mathematically than most.
I still say the common phrasing is "in order of height", (or in "height order"). I'm impressed with Hypsometrical, but as noted "it won't work in colloquial speech" (i.e., you won't be commonly understood). Still, it's a great word :-).
That said it appears the concept of vertical isn't well understood either, but in the context of a picture I suspect many people would understand it. Given the confusion my suggestion generated, it's likely a bad choice unless you understand your target audience well.