Cat training possibility?

Question
This is not a joke or a facetious question.  I am dead serious.   
We have 4 cats, one of whom has epilepsy, and one of whom has a 'sensitive tummy,' and a third of which is a long-hair, and gets hairballs, and HATES/will not lick the hairball remedy stuff. (All are strictly-indoor cats.)
 The ones I've specifcally mentioned have a habit of vomiting on the rug...the epileptic one will sometimes wolf her food after a seizure, and have it bounce right back up...often into the food dish.  But the other ones....hairball, or whatever...
 Is there any way to teach them that if they are going to "urp," to do it in the litter box, or at least on a hard surface floor, instead of the carpet?   The problem is trying to catch them at it!  It will happen when we are not home, or overnight,....seldom right in front of us.  If we DO see them, we snatch them up and toss them into the bathroom & shut the door...
  We "inherited" this carpet when we bought the house, and it needs to last a few more years before we can afford to get new flooring, probably hardwood instead...for just this reason!

Answer
Thanks for your feedback!  I always appreciate when someone writes me something to let me know how I'm doing.  :o)

So, here's the advice I've gotten so far in response.  Every reply so far has been to the effect of the fact that they've never tried, nor think it would be successful, since it's something a kitty does on a kind of spur-of-the-moment basis, not to mention when you approach a kitty while in the middle of doing as such, they usually run off, so it's next to impossible to do anything except try to leave them alone.  

I've about half the time successfully been able to slip a piece of paper (regular letter-sized) under them right when they actually get it up (and not before...they're busy enough doing it then that they don't notice you doing it) to catch it before it hits the carpet, but since you've mentioned that it happens more when you're not home or are asleep, I thought I would give you the advice I got about cleaners that wouldn't stain.

One thing I read in a reply was that when the lady was feeding her kitty Meow Mix (which has red dye in it) the "urp" would stain the carpet, but when she changed her kitty to a food that is without dye, she no longer had the staining problem

Another lady suggested (and it was actually my thought as well initially, but I wanted to check before saying it) the following:

"I have a very light beige carpet and I definitly need to be careful to use a cleaner that doesn't stain my carpet.

The Enzyme cleaner I use is called Pet-Zyme which I get from Pet Smart. It's been the only one I ever used because it works.

I also use a carpet cleaning machine at least once a year and I just put some bissel solution in it- that's safe for animals and helps get out stains that animals leave behind. It seems to work well to get the stuff out that you can't see very well."

My first thought was also an enzyme-based cleaner, that way you can get it at the bacterial level, but I've never myself used as such.  I currently have hardwood floors and tile in the house, so I don't have much to lend about carpet cleaners.  (There were actually two ladies that suggested an enzyme-based cleaner, now that I'm reading every response to give you other suggestions.)

Here are some other suggestions:

"I just clean my carpet after an "urp" with Target carpet cleaner. It sprays on and wipes up with a rag. I have not had any discoloring or bleaching with this product as I have with others."

and another...

"I know one of those martha-stewart types who uses all-purpose flour to clean up most of it (kids or dogs or cats), cause it dries it up and sort of makes it clump. Sort of like your janitor in grade school putting sawdust on the mess.

Then she'd get it with a paper towel and wait for it to dry and then vacuum it, and stain left was gotten with Fantastik (also my mother's old standby, but you can use any cleaner you'd like-- my favorite is the Resolve carpet foam and upholstery cleaner, cause you don't have to scrub it in with a toothbrush as much as other stuff)"

I hope those help!  I can't really vouch for the flour idea...but you can give it a shot, if you like!  Never know...sometimes the weirdest-sounding things work like a charm!  :o)

Either way, let me know how it works out, and which one works best for you, so I can tell others in the future.  :o)

Hugs!!  :oD