Vegan mayonnaise never emulsifies

Starting from the basics: Mayonnaise, as you know, is a combination of water-based liquids, water-soluble ingredients, and lipids (fats/oils). Since water and lipids are immiscible, that makes mayonnaise an emulsion.

Because the droplets (of fat) suspended in an emulsion are not actually dissolved, the properties of that emulsion depend entirely on the size of those droplets and their dispersion. The most likely reason that your mayonnaise tasted like oil is that it actually was pure oil in spots.

The technical term for this is flocculation.

(source: Cube Cola)

This is probably what happened to you - it's possible that if you had really poor dispersion, you might have even been closer to the "coalescence" stage.

To use a more tangible example, consider what happens when you dissolve flour or corn starch in cold water, then heat it. The starch gelatinizes and you end up with a fairly uniform, thick paste. Now think of what happens if you toss it into hot water; you'll tend to end up with something that isn't uniform, instead you'll end up with big globs of cooked flour floating around in thin, cloudy water.

Keep in mind that the chemistry is completely different with an emulsion - in fact, there technically is no chemistry happening with an emulsion until emulsifiers come into the picture - but the concept is the same. You might not be able to see those globs of oil floating around in the water as well as you can see the globs of flour, but if you didn't get proper dispersion and suspension, they're there, and they will taste exactly how you'd expect a glob of pure oil to taste.

Traditional mayonnaise uses raw egg yolk (containing lecithin) and mustard (containing mucilage), both of which act as emulsifiers. These are called "emulsifiers" mainly because they help the emulsion to stay stable, which is why the store-bought mayonnaise doesn't separate (it also probably has a few extra additives). However, they aren't all that helpful for getting that initial dispersion; the most efficient way to do that is to let small drops of oil into a liquid that is being constantly and uniformly agitated.

You can do this by stirring, but an even better way is to use an immersion blender with emulsifying blade. Note that this is not the flat aerating blade that is often confused with the emulsifying blade, nor is it the star-shaped liquefying blade that is the default on most sticks and many manufacturers confusingly call an "emulsifying" blade. The one you want looks a bit like a hubcap; it's flat with several slits or holes and is sometimes also called a "smoothie blade" or "whisk blade":

or

(the one I'm talking about is the bottom left)

These things are perfect for preparations like mayonnaise, but if you don't have one, you can get halfway decent results with a wire whisk. You'll just need to use a lot of elbow grease.

If you get really good dispersion, and use sufficient emulsifiers such that the emulsion doesn't separate too fast, then I promise you, your mayo won't have that "fatty" taste and it will be 1000 times better than the store-bought goop.

You might want to try it by hand with a whisk first, rather than going straight to a blender, though a blender will work fine (though I've only done it with a standing blender, not an immersion blender) The key I've found is the very slow addition of the oil to the beaten egg yolks. (only yolks, no white). You might want to taste your oil to make sure that it isn't rancid, which might be the cause of your "stinky" issue. It does take some practice to pour the oil in slowly while you are beating the yolks, but you can add more oil at a time as your emulsion gets more stable.

I've found that it's not so hard to do by hand, and that you will get a real sense of accomplishment once you've done it.