Eggs are emulsions of fat and water themselves, lecithin and cholesterol being the only emulsifying agents.
Mustard is a good emulsifying agent, so I use some.
I use yolks, a neutral oil and all in a jug with a stick blender.
2 minutes, nary a hitch.
I use sunflower oil, but if I want EVOO flavour I whisk that in later in a bowl.
Hard water will make a difference Jason.