A pretty simple process that seems to work for me.
1. Vacuum thoroughly. If you have access to an air compressor, blow out as much of the dirt as you can. The air works great under the seats.
2. Treat any stains. I happen to use tar remover for any tar or road oil stains. A Product called Kids N Pets stain remover works for most others.
3. Apply some carpet cleaner, but don't saturate the carpet. It just needs to be damp, not wet. I have used Folex, Prestone, Blue Coral and Meg's APC+ and have had good luck with any/all of them.
4. Agitate. I use a medium bristle brush. If I can't get into an area with the brush, I use a cotton terry towel.
5. Since I don't get the carpet wet, I just let it air dry.
6. After it's dry, vacuum thoroughly again to pick up the stuff the carpet shampoo has brought to the top of the fibers.
You really don't need a wet vac, but you do need a good vacuum.
My "Dry" vac is the Metro Vac-N-Blo and it will pick carpeted mats up from the floor of the vehicle.
Charles