I have been exactly like your DD. I'm over it now but I went through phases like that and it was hell (and probably hell for my parents)
I had therapy numerous times where they tried to see what in my past had caused this and couldn't understand that I was just terrified of vomit or the sensation of nausea or vomiting, to the point I would rather die (but I wasn't suicidal). I later learned that the sense of impending doom that gave rise to the phobia is a neurological reaction to certain states of illness, so i was right all along, I just felt terrified and there was no underlying reason.
In the end, over the course of my life what eventually helped was a very calm doctor who was with me when I had a stomach bug and remained totally calm. Even though I was in a panic. That didn't cure the phobia but it seemed to make me think that there was another way to deal with nausea and vomiting. I knew that already but having it modelled to me was a stronger thing. He also gave me an injection to stop the nausea and vomiting and gave me buccastem tablets to take the next day. I could see how these worked, which meant I was starting to sense there was a way to control the thing I feared.
Then I had this really kind nurse who told me that I should carry around Buccastem in my bag. Not safe for a child!! I just want you to know that this was the turning point for me. She knew I wouldn't take them unless definitely ill and that wouldn't be very often obviously. With a child this is not safe or appropriate, but I'm wondering if this could be discussed with a doctor to see if there are any safe methods of her gaining some control over nausea and vomiting, the thing she fears.
Some might say that it's not nausea, vomiting or seeing people sick that needs to be controlled, it's her anxiety. I'm just saying this way around didn't work for me. I had to feel better, which meant first being exposed to the thing I feared (getting a stomach bug and being sick) and then EXPERIENCING that it was ok because the medicene given controlled the thing I feared.
In other words I had to experience the thing I feared and then realise I could cope with it.
I got another bug some years later but I had my buccastem and it was enough to make me feel better and stop vomiting.
Since then I was sick after getting a nasty skin infection, took the buccastem, which helped but it didn't stop me vomiting. It still mitigated it. I realised I was no longer terrified of it.
I hope some of this will help you and your husband understand things with your daughter. My case might not be the same but I've been in a very similar situation so can explain from that point of view. I also know that a CBT trained therapist would likely be very helpful (and if not, find another one).
You're welcome to private message me if you want. The crux of it is that she fears something and she fears she would not survive it or cope (including if she witnessed it in someone else). The solution is to experience it and experiencing coping (not just practically, but emotionally). How you get that probably requires a chat with your GP and / or a CBT therapist.