I have a cake recipe which calls for 1 1/3 cup of vegetable oil. What are the essential properties of vegetable oil in baking? What changes would be expected if I were to substitute peanut or canola oil for the vegetable oil?
Peanut and canola are vegetable oils. Vegetable oil is a catch-all term, it's not asking for a product labelled "vegetable oil" although you can buy it in the store. Oils labelled vegetable oil are blends and can be any proportion of things like canola (rapeseed), peanut, corn, sunflower, etc. Just don't use Olive oil, it is a vegetable oil but it won't be good for cakes!
Different vegetable oils will give different properties although most will give you about the same result. Canola is flavorless and relatively light while peanut is a bit richer and, well, peanut-y. The differences in your cake will be subtle, however.
I use extra virgin olive oil in cakes all the time because that's usually all I have on hand it it works just fine. Never tasted a difference and never had anyone else taste a difference. I expect it'd be the same with peanut or canola, which don't even have a strong fragrance like olive oil.