Can an emulsion enter a cell?
Under the right conditions, emulsions of lipids (fatty compounds) and water can cross cell membranes. If the emulsion is prepared correctly, a structure known as a liposome forms, which is essentially a "bubble" with a layer of fatty molecules on the outside and water at the center. This has been studied intensely in biotechnology, because such liposomes can be used to carry DNA inside (DNA is water-soluble) and then be used to deliver DNA to cells, as in gene therapy. 

Typically, oil, which contains triglycerides, is not used for this though, but another compound known as a phospholipid. This is also a "fatty" substance, but unlike triglycerides, one end of the molecule is polar, which helps it interface with water. Phospholipids are the main ingredient in cell membranes, and therefore liposomes made of this stuff can interact with cell membranes. Essentially, the liposome can "merge" with the cell membrane in such a way that the water inside it (and any other contents) ends up inside the cell. For an example of how this technique is used, see this article.

As for the cooking-related question, I would say oil-water emulsions found in foodstuffs are unlikely to cross cell membranes. They contain mostly triglycerides which are not so effective at this, and they are probably not prepared in the right conditions (oil/water concentrations, etc) for liposomes to form.

Also, even if there would be liposomes in some foods that could cross cell membranes, foodstuffs do not actually get into contact with cell membranes in the digestive system, until they are broken down into smaller (harmless) molecules. In the mouth, esophagus, stomach, etc, tissues that come into contact with foods are heavily protected by layers of mucus to prevent cells getting "attacked" by all the weird biomolecules (and organisms) that we eat.

Finally, various molecules from food don't actually enter the cells of taste buds; instead, they are sensed by taste receptors present on the outside of the cells. (Well, the sour taste sensor involves transport of small ions, but not larger molecules.) So you should probably revise you answer on Cooking SE.