If a child writes on her hand will she get ink poisoning and vomit? Her hand is red and she's vomiting. She doesn't have fever or the poops.

No. No, she will not get ink poisoning. That's a bit of an "old wives' tale". That said, if she's irritated her hand, it may be a little red. I highly doubt the vomiting has anything to do with it, and is likely another issue. If that persists, i'd get her evaluated.
No. She will not get ink poisoning. Vmiting can happen due to other reasons. How is her apetite? Is she stii able to drink fluids well and eating well. If she does not want to drink/can't drink or eat- you need to bring her to pediatrician or er.