The absolute easiest oil to produce at home with absolute minimal effort is going to be sunflower oil. Sunflowers practically take care of themselves once you stake them, and can be grown in almost any climate zone and in abundance. Once they're done, chop the heads off before the birds get to them. Getting an oil press will net you all the cooking oil you could want for a year shy of needing large amounts for deep fat frying. About 1.5 Kg of seeds gets you about 1L of oil. Make sure you plant a high-oil variety, usually the stuff advertised as bird seed grade "Black Oil Sunflower seeds" is what you want.
The best part is, sunflower oil is pretty tasty in dressings, has a high smoke point, and is ~~super~~ relatively healthy for you as far as oils go.
Extra bonus: You will attract birds (good or bad thing) and you will attract pollinators like bees.
Would you be up for making your own out of organic peanuts using an oil press? You know, for the sake of TRADITION!
TIL http://www.survivopedia.com/how-to-make-vegetable-oil/
https://www.amazon.com/Piteba-Nut-Seed-Expeller-press/dp/B004H2SDTM
I'd probably render a duck down to make oil.