Lye is what causes the chemical reaction that turns oils into soap.
Why are you reluctant to use lye? Is it because it is caustic? If so, I can assure you that if you use normal precautions like you would when handling a big, full bottle of bleach, you will be just fine. Were rubber gloves and eye protection and make your soap when the children are in school or at Grandma's. It cleans up with a rinsing of cold water to wash it down the sink.
Lye used to be sold as drain opener....I used to buy it in my local grocery store and use that to make soap.
FWIW I can't get lye locally anymore due to it being a main ingredient for making meth... :/ There are several soapmaking supply companies that you can buy it online but you have to fill out a waiver or purchase with base oils and only allowed to buy in smaller quantities. But it's still PLENTY for soapmaking. I was nervous the first time I made soap too for the lye factor but it really wasn't a problem at all. I was worried about fumes and opened every window in that half of the house but I actually didn't smell a thing!