I’m not really into cooking, but I’m familiar enough with chemistry to know that you need an emulsifier of some sort.
Mustard, as recommended by the only other commenter at time of writing, seems to be common for salad dressings.
Another recommendation would be to simply try to prepare your vinegarette closer to the time that you’ll actually need it.
Egg yolk is another common emulsifier, though it depends on if eggs are safe to eat raw where you are: as an American, I was shocked to learn that it’s actually really common in Japan to break a raw egg yolk over rice dishes to add protein and flavor, because raw egg yolk is disgusting to me.