Any kind of material which has greater density than water will sink. Oil is lighter than water so, that means any amount of oil will float in water. That means oil cannot sink in water just like a submarine.
The problem is how to make oil dense than water or how to decrease the density of water, to allow the oil to sink.
In this case, you can make use of a kind of oil that has a greater density than water. Some oils may contain suspended solids which can make it heavier than water. Bigger objects with heavier weight will certainly have an increased density preventing them to float in water. In some cases, heat may be applied to let oil sink in water.
When some objects are heated, their density decreases, but the rates vary accordingly. A kind of oil which has a low coefficient of expansion may stay close to its original value when the density of water is reduced in heating. If the low density of water is obtained before it boils, you can be sure that the oil will sink.
If something floats or sinks in water, it has something to do with density. A stone is heavier than a piece of styrofoam of the same size. The density of oil is the same as the styrofoam, and the density of water is like the stone.
In the law of buoyancy, we talk about the buoyancy force that keeps an object float and makes another object sink. Objects that are less dense are moved in upward direction and those that are denser are moved downward. Since oil is on the same feature as a styrofoam that is less dense, oil is therefore moved in an upward direction, while the water which is denser is moved downward.
Therefore, in order to allow the oil to sink in water, you have to make its density greater to the point that it is more dense than water. However, that thing is very difficult to achieve in liquid form because you cannot squeeze it just like a styrofoam which you can smash altogether like a rock to increase its density.
So, to make it possible for the oil to sink, you have to mix it up with something solid. This is the same case with seawater being more dense than fresh water since the weight of the salt adds up to the weight of the water. The only way to make oil sink in water is by pushing the oil into the water with your hands, or mixing it up with another element to make it dense.
In the case of vegetable oil mixed with water, if you shake it up inside a sealed bottle, it appears to be mixed, but actually it does not. They just separate with each other according to density, and the oil will float on top of the water. Anything which is less dense than any fluid will absolutely float on top of it.