Depending on the volume and person, yes consuming oil straight up can cause vomiting…
Reddit: *What would happen if I drank a cup of vegetable oil?*
“A cup? That’d go right through ya, provided you didn’t promptly throw it back up. Your body can’t process much of the fat (though yes, it would process some) and much like how Olestra works in fat free chips, the oils got to go somewhere and there’s only two ways out.”
In regards to a not-quite-human animal throwing up after drinking oil:
“Cooking oil as you have found out can indeed cause vomiting and diarrhea. You may want to try some burnt toast and see if it absorbs some of the excess oil. However, greasy, fatty foods can cause pancreatitis. Symptoms of pancreatitis vary but include vomiting, diarrhea, anorexia, weakness or collapse, dehydration, shock, fever, depression, and upper abdominal pain.” (Jane Lefler, former Vet. Asst)
Also, lead-based paint chips are totally different. They settle in the stomach. From what I’ve learned, children would gnaw or suck on them more than eat them like a bag of Lay’s.