I got sick, but only once, and only right before I delivered her. It's one of those situations where it was the last thing on my mind. She was coming out in a matter of minutes, I puked, then I pushed, then there she was.
I know fears are usually irrational, so there's no talking you out of being scared. Just know that IF it happens to you, it will happen quickly and it will be over just as quickly, and you'll have better things to focus on.