Did any chemical reaction occur between the coconut oil and H2O2 to make a new compound?
Most likely, no chemical reaction occurred.
Whether any reaction occurred depends on the conditions you used when mixing the oil and hydrogen peroxide. How long did they mix for? What temperature was the mixture at? Did you add anything else to the mixture?
Hydrogen peroxide is used in the production of epoxidized vegetable oils. However, the reaction is performed for prolonged times at elevated temperatures with concentrated hydrogen peroxide in the presence of a catalyst.
I am guessing that you just poured the two together at room temp, then soon after used the mixture to wash your hands. Compared to the commercial reaction, you are missing time, heat, and a catalyst, so it is very unlikely that any reaction occurred.
My guess would be, since Coconut Oil is mainly lauric acid, small quantities of peroxylauric acid may have been produced?
Oils are composed of triglyceride esters of fatty acids, not fatty acids themselves. So you would not see any peroxylauric acid.
It definitely felt less greasy than pure coconut oil, left my hands feeling clean and soft :)
It felt less greasy because it was less greasy: you added water to it (3% hydrogen peroxide is $\ce{H2O2}$ in water). You probably made an emulsion of oil and water and used that to wash your hands (kind of like using soap).
