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.
You probably would not notice much difference substituting peanut or canola oil for "vegetable oil", because of the amount of sugar in a cake. On the other hand if you substitute with extra virgin olive oil you will notice a difference, not so much in the flavour, but in the texture. Unlike the other oils mentioned EVOO contains natural mono- and di-glycerides, which are emulsifiers which will give the cake a fine, dense crumb, and help it to hang onto moisture, so that it remains fresh longer. There is a long tradition of using EVOO instead of butter in cakes in places like Italy and France.
As an aside, in the case of pie crusts, which do not contain sugar, I have noticed a flavour difference between oils (if you have not tried making pie crust with oil, do yourself a favour and give it a try; they are super easy). Avocado oil has a slight fruity flavour, and canola had a slight fishy flavour (maybe because it is high in Omega 6) which I find unpleasant. My favourite is peanut oil which has a rich taste (highest in saturated fat) and seems the closest to butter. Unfortunately I did not include EVOO in my side-by-side testing but I am guessing it would produce a very tender crust similar to ones made with shortening, since they both contain mono- and diglycerides.