After my post last week on replacing eggs, I got a lot of questions about how to replace oil as well. While vegetable oil is plant-based, I don't use it when cooking or baking. I did 2 separate posts last year on replacing fat when cooking and baking as well.
Oil is most commonly used on the stove for sauteing garlic and onions and frying food. I primarily use water (sometimes vegetable broth) for sauteing. Start with a little liquid, about 1/4 cup, in a pan over high heat until the ingredients are cooked and most of the liquid has evaporated. You may need to add more water when cooking to prevent sticking and burning before they are cooked.
As for frying, I've found that I can bake most things and still have that fried, but not greasy, taste. The Baked Onion Rings in HHC (p. 176) are a great example of this. For altering an existing recipe from frying to baking, preheat the oven to 350F. Line a cookie sheet with parchment paper and then bake the food for 10 minutes, flip and bake for 10 more minutes. Continue this cycle until the food is cooked and has a nice crisp outer crust.
Baking is another place that oil creeps in, whether it muffins, breads or cookies. My usual go to replacer is applesauce, it's great for cupcakes and muffins, but only good for soft cookie recipes. My next favorite substitute is canned pumpkin. Although this can leave a slight pumpkin flavor and orange color to the muffins or cake, sometimes this is a plus like with the Chocolate Zucchini Muffins.
Of course there are other types of fat besides oil that come up when baking and cooking, such as heavy cream, avocado or coconut for example, that you may also want to replace. Here are 2 handy pdfs of how to replace fat - such as oil, butter or cream - when cooking and when baking.