I have three in my household that currently have bad coughs (ages 3, 16, & 20). There is no fever or congestion or runny noses or sneezing. Just a hacking cough that sounds like they will vomit, but they don't. What could this be?

We can't say online of course, but I and others I know are familiar with a cough of this sort, and it really was just a cold virus with a variation. In my case there was a very glue-like post nasal drip with no runny nose, and that was making me cough. You have to drink a TON of fluids, gargle once an hour with salt water, and suck on menthol cough drops.
But if any of the mucous has gone down into the bronchials, you may still have a cough. If it lasts more than 3 weeks, it's turned bacterial (vs viral) and you need antibiotics. But it's fruitless to try to treat a virus with antibiotics.

Try eliminating sugars/sweets, artificial sweeteners and dairy.

Are they vaccinated against whooping cough?

I had it several years ago and it's a cough with no runny nose/fever/mucus.
It didn't ave the whooping sound in the beginning but it's definitely very strong and might cause you to vomit.
Went to a regular doctor who didn't realize what it was and who tried to treat it with cough drops for a week then with steroids and then it was too late to be actually treated for what it was (whooping cough is a bacterial infection that can be treated with antibiotics in the first 2/3 weeks). It then turned very bad and I couldn't breath properly for several weeks and it took several months to heal.

If your children suffer from this cough for a week or more, definitely go see a doctor.

They have been finding that nost outbreaks of whooping cough the people Were vaccinated for it.