The key to removing vomit from leather furniture is to act fast. Leather is a porous material that will absorb the stain and the smell of vomit and other liquids if they are not removed quickly. Attack the problem as soon as you discover it or the stain could become permanent.
Wipe the bulk of the vomit off the furniture using paper towels or disposable rags, and throw them away. Use a spoon or spatula if necessary to remove any solid material that remains.
Find the right cleanser for your leather furniture. If you are dealing with suede or Nubuck leather, use a cleanser specifically designed for these types of fabrics. For other leather surfaces, a basic leather cleaner will do the trick, as will soda water and distilled water. La-Z-Boy recommends mixing a 50/50 solution of distilled water and mild soap.
Dampen a white, lint-free cloth with your chosen cleaning solution. Squeeze the towel out so that it is damp but not saturated. Remove the stain with the rag, rubbing with the natural grain of the leather. Do not rub in circles or against the grain. If the stain is stubborn, use an old toothbrush to gently wipe it away, again moving with the grain.
If the furniture smells clean, there's nothing left to do. But if you can still smell vomit, sprinkle the leather with baking soda and let it sit overnight to absorb the odor. In the morning, sweep the baking soda into a dustpan or vacuum it off the couch. Repeat if the odor still remains.