Er, I don't think emetophobia in of itself is inherited. The underlying anxiety that can cause a specific phobia, yes. Undoubtedly, I'm an anxious person because it runs in the family. No one in my family is emetophobic in the slightest, except for the fact my dad doesn't like throwing up and would rather endure nausea, but he isn't afraid of it at all. My dad has GAD, that's where the anxiety came from.

My mom and her mother both have severe specific phobias, which is where I'm sure I got emet from.

Phobias develop because of a combination of inherited anxiety and a traumatic experience (of which there are varying degrees) involving the trigger when your anxiety was high, generally - at least that's what I've concluded after doing some research on it. There is some good evidence that you can be born more predisposed to developing a phobia if someone in your life has one, but I doubt it's for specific ones, like a specific arachnophobia gene. Otherwise, I would be petrified of airplanes and heights instead of vomit, which happen to be two things I have zero fear of.