What influence does the properties of fat in vegetable oil affect emulsification of oil?
Fatty acids are not the emulsifiers here. Long chain fatty acids are excellent for emulsification, if they are deprotonated.  But then they would be called “fatty acid salts”, and their flavor would be soapy - bitter.  If they're neutral fatty acids, they're not ionized enough to retain a sphere of water around the micelle, and block aggregation of the droplets of fat in an emulsion. 

Note that the term “fatty acid” is often misused. It is used to describe structural components of triglyceride fats, but in that form they are esterified, and not in their acid forms.  They only become free fatty acids after the saponification (breakdown) of a fat, which can happen when cooking fats at high temperature.  It's convenient to talk about them as fatty acids when they're still inside triglycerides because of their roles in nutrition (saturated vs. unsaturated vs. polyunsaturated, etc.).

An emulsifying agent like lecithin has two non-polar (electrically neutral), lipophilic (fat soluble) “fatty acid” (but still esterified) hydrocarbon chains, and a polar head consisting of a phosphate/choline (-/+) ionic structure that is very hydrophilic (soluble in water) due to having both positive and negative standing charges.  Because of its net electrically neutral structure, the flavor is mild and fatty, not soapy.  Phospholipids like lecithin are available in all cells, since they form the lipid bilayer that defines each cell.

Neither vinegar nor lemon juice would act as an effective emulsifying agent.  Acetic acid, citric acid, etc., are all too water soluble, and wouldn't stick to the surface of the fat droplets in an emulsion. Rather, components of the soy milk, particularly soy lecithin, would fill that role in this recipe for vegan mayonnaise. In traditional mayonnaise, of course, egg yolks are a rich source of lecithin.  The vinegar has little effect other than flavor.

The degree of unsaturation in the non-polar tails of a phospholipid would not significantly affect the stability of an emulsion once formed. Unsaturation of fatty acid esters in triglycerides does affect the texture of a fat, and how it varies with temperature.