golly gosh, what do they eat? Just meat and dairy?
Could they mean plant oils are the problem, extreme nut issues? If that's it, straight jojoba, it's a plant wax rather than an oil, almost the same composition as the skin's own sebum, or aloe gel.
If jojoba is out, obvious idea is the ol' petroleum products, mineral oil, KY jelly etc....
less obvious ideas clay as a dry medium, laponite gel would be a good option (it's a mineral gel, not petroleum based)