Is it appropriate/accepted for a PhD student to organize/coordinate a special session at a conference?
I have done so, and the participants weren't troubled by it at all. As a PhD student you are supposed to become an expert on your (very narrow) topic, so near the end you are supposed to know more than most about that one topic than others in your field. 

Susceptability to status differs quite a bit from country to country and discipline to discipline, so whether this works in your field depends. I am Dutch, but the conference was an international one, and my discipline is sociology.