Jesus...obviously no one responding yet has owned a cat, otherwise there would be some useful advice on cleaning up puke.
If the puke isn't that bad, I'd just get a scrub brush and some spray-on carpet cleaner like Resolve. Spray the cleaner on the carpet, let it sit for a few minutes, then scrub the carpet really good to break the puke up and get it out of the carpet. Get the vacuum out and go over that section of the carpet to pick up any loose bits of puke the brush didn't pick up, and to "smooth" the carpet back out (the fibers will be all flattened and look weird after you finish scrubbing). I've done this a million times over the years since cats like to puke in the middle of the fscking carpet (and move around, so the stuff gets spread over as large an area as possible, evidently). Works great and even the puke that wasn't discovered for a while (and dried up) will come out pretty easily this way.
If it's really that bad, then go to the grocery store and rent a Rug Doctor or something for an hour and go to town.