According to the documentary that I watched last night, there is no magma chamber, per se'. The Yellowstone Caldera is connected to the molten core from a large outcropping of magma caused by gases being released by the magma as it churns in the planets core. A huge magma tube (believed to be more than 5 miles across in some points) channels the molten rock and gases towards the surface. This fuels all of Yellowstone's trademark attractions. A weakness in the North American plate combined with this outcropping of magma caused this huge volcano to form several million years ago. The Yellowstone Caldera is credited with causing at least one of the ice ages. Still relatively recent in the geological time frame of our Earth. Hope this answers your question.