Welcome, Carolynkc.
Lauric acid is a fatty acid in oils; it is not only from Coconut Oil. Not only can it be found in animal fats, plant fats, but also in breast milk.
I am wondering if you also have a sensitivity to Lauric acid, specifically? If that is the case, and you want to avoid oils with Lauric acid, you can identify those easily by sorting the oils in SoapCalc (
here). To sort for lauric acid, choose Lauric Acid. See below:
View attachment 50053
But as mentioned by others, a laundry soap made without lauric acid may not clean as well, so if you choose to make one with only oils with 0% lauric acid, I suggest making a small batch to begin with so you can evaluate it's performance.
I have read of people who make laundry soap with tallow soap, which has only 2% lauric acid; or you could make it with lard, which has no lauric acid.