@EmetophobicElle solidarity, I have this too - I could have written your post, and have been scrolling incredulously at all the posters saying "it's just sick" etc. 
I come out in a cold sweat if I read on a local Facebook page that "there's a bug doing the rounds", or, worse, if I hear there's a vomiting bug doing the rounds at school (I try not to listen to other parents' chatter at drop-off/pick-up, as I am paranoid that everyone is talking about how their DC have been recently sick).
Just like @bumblenbean , after I hear about a bug or feel that we have been in a situation where we might have picked something up, I then calculate how long since the potential exposure and work out when I feel "safe" that no-one has caught anything. A lot of worrying goes on during that possible 'incubation period', and hypervigilance for any symptoms in me or another family member (silently!! - I do try to keep all my worry in my head, I am very conscious about not wanting to pass this anxiety on to my DC).
I have a lot of cleaning rituals that I do whenever we return from anywhere public (and especially when DC come back from school 
) - although I think they are proportionate and sensible (of course I do! ). I too am very avoidant - eg I have only ever been to soft play once, for me that is the worst place (especially as it seems kids often do get vomiting bugs from soft play).
I will avoid public toilets when out and about, and try and hold on for as long as possible, for fear that people have been being sick in them and I will catch something. I've had to temper this somewhat since DC toilet trained and of course if they need to go when out, I have to take them! I suppose that's a bit like exposure therapy? (I try and tell myself in the hours afterwards "look, we went to a public toilet today and everyone is still fine, no-one caught anything").
I also get very VERY angry at people disregarding the "48hr rule" and going out and about whilst contagious with a vomiting bug. Especially sending kids to school whilst infectious. WHY inflict this avoidable misery on the rest of us? I really wish public health gave more prominence to vomiting bugs, and especially noro - I think it should be a notifiable disease (it is in other countries).
For me, my phobia isn't the act of being sick (or helping a family member who is being sick) that is the issue - as pp say that is unpleasant but you just get on with it. I was sick a lot in my pregnancies. It's a) the cleaning - again not the act of cleaning, just that takes ages to get through all the washing/drying, and I panic about the duvet etc getting sick on (hard to wash!) - and b) the disruption to plans. I often have a lot going on, and the thought of having to cancel plans and let people down as a vomiting bug makes its way around the family is the worst thing, I think.
@Somethingsnappy Does anyone who has this phobia have any insight into what caused theirs?
I have wondered about this. I have vivid memories of a school holiday ruined by a vomiting bug when I was in primary (went through whole family in sequence). It stopped us going on a planned holiday to my grandparents' house, which I used to love. I remember it being a really quiet and boring holiday. So I think that's the root of it, and it ties in with "potential disruption to plans" being a big part of my phobia.
However, I wouldn't say I was phobic right from that point - I don't remember being avoidant or hypervigilant about sickness bugs until I became a parent.
I'm working on my emetophobia with my therapist and really hoping it will reduce the time and mental effort that I expend on this. One strategy that I have found helpful is to bring my focus back to the present, like mindfulness - "no-one is being sick right now, if someone is sick in the next second/minute/hour/day I will deal with it then, but in this moment everything is OK". And repeat.
