Which of these organizations' flags is in the picture?

It is very similar to a Caduceus, the staff carried by Hermes which has two intertwining snakes. Its similarity to the staff of Asclepius has lead to the adoption of the caduceus as a symbol of physicians and medicine. I believe it’s the official emblem of the U.S. Medical Corp.

Holly, that's true, but only by mistake. Asclepius was the god of medicine and healing. Hermes was the messenger if the gods, a smooth-talking con artist, i.e., the god of used-car salesmen.