My 13 week old baby has been vomiting after feeding, assumed it's to do with his teething, today it seems to worse as he is crying none stop and vomit

Doubt teething: A kid at that age is more likely suffering from reflux ( the return of food to the mouth from weak stomach entrance). This issue is possible in all infants & peaks in frequency at 3-4 mo.The stomach acid triggers pain (heartburn). There are meds that can help. Sometimes thickening feeds & feeding less but more often helps. I would discuss it with his doc.

Not related: Although rarely a child's born with teeth already erupted, most children do not begin to teeth until about 6 months. Biggest concern is dehydration. Important that you call his Pediatrician immediately.