I think it is difficult if not impossible to find such an oil-soluble protein. Evolution has built proteins using water as an environment. Therefore, all proteins take on a water-dependent structure. Only a trans-membrane domain could have the characteristics that a protein in oil requires, but I imagine its practical use in food difficult.
What about lipoproteins? Many enzymes, transporters, structural proteins, antigens, adhesins, and toxins are lipoproteins. Examples include plasma lipoprotein particles (HDL, LDL, IDL, VLDL and chylomicrons).