Paper towels work well for blotting.
Carpet bears the brunt of most pet stains. Solid matter such as feces and vomit needs to be picked up immediately so it doesn't dry onto the carpet fibers. Urine needs to be blotted up with clean paper towels until most of the moisture is absorbed to keep it from penetrating into the pad under the carpet.
Liquid stains that have dried need to be rewet with plain water, then blotted up. Urine crystallizes deep inside the fibers when dry. These crystals must be dissolved in order to remove them. Wetting the area then blotting will remove much of the stain and odor.
Use a spray bottle with the vinegar solution.
Vinegar is a stain remover and odor neutralizer, making it one of the most valuable cleaning products in the pantry. It is mild enough to be used on most carpets and fabrics without damaging them, and it doesn't irritate the skin. Vinegar even works to neutralize old stains.
Vinegar is mixed with water in a 50/50 solution. It is best to apply it with a spray bottle so that the mixture evenly coats the stained area, but it can be blotted on with a towel if necessary. Enough should be used to permeate the entire stain, especially in carpets with pads underneath or upholstery. The vinegar is then blotted up, removing most of the visible stain as well as much of the odor.