Vegetable oil is almost always soybean oil. They mean vegetable in the “animal, vegetable, or mineral” sense, not in the “leafy green vegetable” sense. In other words, they mean oil that doesn’t come from an animal.
FTR, peanuts are around 48% oil (by weight). If you want to see it for yourself, throw a bunch of shelled and roasted peanuts (maybe a little salt) in your food processor and process until smooth. Let it sit for a few hours and watch the oil rise to the top. Or, buy some “natural” peanut butter (the kind that doesn’t contain hydrogenated oils in it). The ingredients list will usually just say “peanuts” and the jar will have a layer of oil sitting on top of the peanut butter.
Particularly in the case of cheap brands, peanut butter may be made from defatted peanut meal (a byproduct of the peanut oil extraction process) with some other kind of oil added (oibviously a cheaper one that peanut oil, or the process would be pointless), such as rapeseed (I think this is called canola in the USA) or palm oil.
OK, sure, most vegetable oil is from soy…but what about corn oil? I can go buy a bottle of 100% corn oil. Do corn seeds have that much oil? But aren’t the seeds the kernals? So maybe corn oil comes from a different part of the corn?
All grains contain oil. Corn included. It is in the seed. Specifically, mostly in the germ and the hull and not in the endosperm (the soft, sugary part of sweet corn that tastes so good). Since corn used to produce corn meal or grits is hulled and degerminated first, producers have all of these germs and hulls left over that they can squeeze to get the oil out. Hybrid producers are also working on high oil corn varieties (mostly for llivestock feed).
Peanuts have a lot of oil in them. You’re thinking of dried peanuts, I think. You make peanut oil pretty much the same way you make olive oil: by squeezing peanuts.