Even vinegar might be too strong for this. My soap recipe is extremely gentle on the skin. People who have not been able to use anything else have been able to use my soap without adverse reaction. Coconut oil is very popular in soaps today. I don't use coconut oil because it makes a rather harsh soap for skin.
Olive oil makes very fine bubbles and is very good for the skin.
Concerning soap lather, I've had no complaints from people who use my soap. The fine bubbles from the olive oil are enough. Four hundred forty grams of finely ground comfrey root in a 14 lb batch of soap stops poison ivy itch. Old timers used to put pine tree rosin in their soap to help it suds. Soap with pine tar also makes an excellent pet soap.