Sears used to sell the "16927 carpet cleaning attachment set" for 54.98 (in the 1997-98 Craftsman tools catalog) which was basically an attachment set and 40 ft of braided vinyly hose, floor and upholstery tools, extension wands,water valve, faucet connectors and adapters, etc that basically turned your wet/dry vac into a carpet extractor machine.
Unfortunately it seems they no longer sell this but I have that catalog and that is the description... seems like someone ought to make it.
You can get a similar attachment alone (floor/carpet head) from Sears for about 12 bucks and it looks similar to the Rigid tool at Home Depot for about 15 bucks.
A good wet/dry vac works well.... I'm using my old Craftsman to do my carpets because my Bissell all of a sudden won't spray solution anymore (it would and then it won't, LOL) I think it has an air leak in the lines somewhere as I've checked all the "o" rings and other seals... it will shoot a bit and then quit.
SO.... I've just been using a garden sprinkler can for my solution and then water. You can find good home-brew carpet solutions made from Tide/vinegar/ammonia/water or clear dish soap instead of Tide... some people use just vinegar and water if they don't like the chemicals.... for extraction I find a splash of Tide (maybe 3 oz or so) and a glug of vinegar (maybe 3 oz or so) to about a gallon and a half of warm tap water works great. I'm spot treating with Folex and/or Zep high-traffic carpet cleaner but a good substitute for the Folex is 1 quart water, 3 oz vinegar, 3 oz ammonia, 6 oz tide or clear dish soap (actually works better than Folex, seriously)you can leave the ammonia out if you have cats as they will take the odor of the ammonia as urine and to mean it's OK to use the carpet as a cat box, I hear on the 'net.
The Zep has enzymes so I spray a little spot cleaner on the spots (Folex or the home-brew) then mist the worst or the area with a pump-sray bottle.... liightly rub the surface with my hand just to spread it a bit, then soak (sprinkle) the area with the sprinkler can (with the diluted home-brew, mostly water with tide and vinegar), rub by hand again (lightly, just to spread again, then vac it all up....works great.
For general large area cleaning you can use the home-brew and sprinkler can and vac up immediately. I bet you can get good results with a 3-gallon garden pump sprayer if you don't want to get the carpet so soaked but my big wet vac pulls it up pretty good..... remember to let it get some air flow and I use the 6-inch gulper nozzle for most of the cleaning... you can actually pull up the carpet a bit and pull a lot of air through the backing and all. I use the car attachment (similar to the crevice tool) for spot cleaning or smaller areas (like next to walls or door jambs) but the crevice tool might work as well other than being a bit more awkward.
The only problem with soaking with a sprinkler can is you can "wick up" some dirt from the backing but if you're soaking well you'll pull it all out, otherwise you'll have to hit it again. A 3 gallon pump sprayer might let you just wet the nap without soaking all the way to the backing so you can just clean the top which will also allow for faster drying... although I'm not having any problem with it, only taking slightly longer than an extraction machine (although I suspect if I soaked it that much with an extraction machine it would take about the same amount of drying time.... Just make sure that if you have puddling under the carpet, which you can feel "gulping" up the hose/wand, to keep gulping that area until you don't feel the water coursing through the wand anymore in that area (otherwise it will take way to long to dry)... then just keep pulling the gulper attachment along as usual. the top will be damp but will dry in a few hours I use a fan to speed it up, you can also use the a/c to pull water out of the air and speed drying.
I'm using a 2.5 HP, 8 gallon, 2 1/2 inch hose Sears wet/dry vac I got in the '80s... I bet one of the newer 5 HP models would work really well but I'm getting great results.