Worth repeating - being sick with a reg in your mouth is no biggie. In fact it can be a good way to see the fish.I've been sick underwater a number of times. The first time was a bit scary, but I remembered NOT to take the reg out of my mouth. Throwing up in your regulator isn't hard to do. Throw up and then hit the purge button and everything will come out of the reg (and many little fishes will come to feed). Take a breath from your reg and you'll be fine. If you have to throw up again, just follow the same instructions.
Also worth mentioning is that if you have actually been sick, you are more than likely dehydrated to a certain extent so it would be worth having a drink of fresh water or similar prior to diving or ensuring your dive profile is conservative as you will have a higher risk of DCS.