Whatever it is probably isn't contagious.

A lot of times, vomiting is the end of the issue. It gets the toxin(s) out of the system and you're able to settle down. In other words, those family members aren't going to be sick at the time of the flight.

There doesn't seem to be an easy solution. All options are bad, so just stick with the original plan and do your best to not impact other passengers on the flight.