At the station I used to work at in Lexington, KY, the main male anchor had cancer (he has since passed on) and had to use a wastebasket occasionally. I believe that they would just cut to the other anchor to not show it.
The anchor was open about his cancer and we did a bunch of stories about him and the disease.
Now, for myself, I've barfed many times at work and on stories. With the GI troubles that I'm having, it's becoming a regular event. Many times I've pulled over, told the reporter that I'd be back in a minute, and do my business. Wash the mouth out real quick and on with the story.