I assume by 'living organs' you mean human spare parts? To make it short, the way you desribe - private aircraft and PPL pilots - will be NoGo in most countries. For legal permit to plant used human organs in other bodies you have to fulfil quite a lot of bureaucratic checks, i.e. document cool chain, have a license allowing for transporting potential dangerous goods etceteras. As a private pilot running a private aircraft you would have almost no chance to bring all necessary papers, so a doctor can legally implant the organ in most parts of the world.
I remember reading a story lately of someone trying to transport living sheep in Europe - a MontyPythonian story of 2018. Try to find that and imagine what this makes of your idea of human parts.