I haven't played the game but from screenshots the Dont Starve images appear to be hand drawn sketches probably done on a Wacom drawing tablet in photoshop, or maybe hand drawn on paper and scanned in, it's definitely not vector art. It Also looks like 2d in 3d engine which gives it that extra paper cutout effect.
I'm not sure what you mean when you say she needs to do it in vectors, are you confusing Construct with Flash?
Anyway Construct doesn't support vector art directly. How you make the image and in what software doesn't matter as long as it saved in a file format that Construct supports like png or jpg.
She could always draw the character as separate parts like head, body arms and use Spriter http://www.brashmonkey.com/spriter.htm to animate it.