You probably want to use a Pigma pen; Sharpies tend to bleed a lot. I would probably look for a smooth cotton ribbon (if such a thing exists) because the long sides wouldn't ravel; you would just need to tuck the cut ends under when you whipstitch. It's much easier to write legibly on fabric that is smooth.
Is this a private kindergarten? I've heard of marking clothes that are going with a child off to camp, but not to a day kindergarten -- except maybe outerwear and mittens.