Hi there, I'm quite badly emetephobic as well, so I totally understand your worry! I've always been desperate to be a mum but it took me a while to take the plunge due to my phobia.
I'm 10 weeks pregnant and have unfortunately suffered with severe 24/7 nausea but not much throwing up. My mum didn't get morning sickness so I was hoping it was genetic! My granny (mum'a mum) had it quite badly. So sadly, I don't think it's particular genetic
My main recommendation is to make sure your doctor knows and hopefully is understanding of a very real phobia. Mine have been brilliant. Pre-pregnancy we discussed coping techniques and medication risks/benefits. I did my best to cope for a couple weeks when it started but when I actually started being sick, I lost it. Full blown panic attacks, crying all day, I wasn't coping. So I went to my doctor and we agreed medication was the way forwards for my mental health. I'd stopped eating and was boarderline going into hospital with dehydration. So I've been on them ever since. They've barely touched the nausea at all but I've not been sick since. And that's taken away the panic and fear for me massively. I do have days (today is one of them) where I feel quite close to being sick and I cry my eyes out but overall I've been able to get to work most days and get food and drink down me!
I keep everything crossed for you that the non sickness keeps running through your family! Good luck Mrs Vet!