Symbiosis between a large organism and petroleum metabolizing bacteria
Living things operate in water and oil and water don't mix, so it is very difficult for living things to use pure hydrocarbons. Fats used by your body are mostly not pure hydrocarbons. For instance, fatty acids have carboxylic acid group that becomes negatively charged in water at neutral pH, making them much more water soluble than their purely hydrocarbon analogs (however, they still need help from lipoprotein particles to be carried around the body). The carboxylic acid group provides a handle for doing further chemistry as well.
This paper provides a lot of the details, but here's a good quote:
Due to the hydrophobicities and low water solubilities of most petroleum hydrocarbons, the biodegradation rate is generally limited in the environment.
For this reason, with the right selection pressures, I think that large organisms might actually be able to do much better than bacteria at metabolizing petroleum, since they can create internal environments (temperature, pH, oxygen concentration) more suitable for the reactions necessary to break down hydrocarbons than microorganisms.
While it's possible that the large organism could evolve the enzymes for degradation itself, I'm imagining that they might evolve like cows. The cows themselves do not produce the enzymes necessary to digest the tough plant matter that they eat (mostly cellulose and lignin) but create an environment that in which can do it for them. Chewing their cud provides mechanical mixing and the bacteria, protozoa, fungi, and archaea in their rumen convert the cellulose and lignin to fatty acids and other fairly water soluble molecules that can be used directly by the cow.
So, I imagine that a large organism could evolve a kind of stomach populated by petroleum-degrading microbes that provides an optimal environment for the microbes, uses mechanical mixing to expose more oil to these microbes, and helps to efficiently solubulize the oil by secretion of some kind of surfactant.
Why haven't such creatures evolved on Earth? As mentioned in the other answers, maybe there just isn't enough petroleum on Earth for this to be sustainable. However, you could imagine a planet where petroleum is more abundant and produced at a greater rate. Such petroleum cows might evolve there.