I would say i did have a phobia, I even once got of a bus in the middle of no where because I thought someone who was coughing was going to be sick.
I managed to avoid being sick myself almost allmy childhood.
When I was nannying I had to deal with one of the boys one evening, I was just about managing and the mum came home as soon as i heard the door opening that was it, I ran outside I could cope no more. I think its because of nescesity with kids that you find ways of coping.
My 2 kids have had numerous incidents thankfully they travel well. I find I can cope not to bad with them and would say it is not a phobia anymore.
On a practical note i recommend pampers care/ change mats. Any sign of them being under the weather and i tuck one of these under them. Any accidents from either end and you just roll it up and bin it. I have seen me with 3 or 4 of them down the whole length of the sofa.
In the car keep them looking out of the window and don't let them read or even play with toys that stop them looking out of the window. lots of eyespy, make sure you have fresh air circulating and not too warm, stop lots to allow a chance to freshen up and carry care mats, bags, wipes, hankies and anti bac spray so if anything does happen you know you've got stuff to deal with it.
I would recommend seeking advise from your GP too. I never got that far as I seemed to desensitise myself when the kids were baby and baby sick wasn't to bad. Good luck and thoughts to you .