It's easy to get lost
in the term "organic" as it has developed new and vague meanings. All soaps are long organic molecules that have an affinity for water on one end and an affinity for lipids, oils, waxes, grease, (non-polar molecules) on their other end; therefore, they allow water to mix with oily things and then rinse away.
So, if you want organic soap, it's all organic!
Now, if you define organic as being something that is not harmful to the environment
you still have many choices, but with this use the term, "organic" loses any clear definition and is not a scientific term. Your best choices here are to look for soap without phosphates in the ingredients.
How about Johnson's Baby Shampoo?