While I wasn't able to find much information on the seeds of the Phantom Orchid specifically, there is some interesting data on Cephalanthera in general. Orchid seeds are some of the smallest amongst angiosperms. The common explanation behind this is that the dust-like seeds are easily wind-borne, and thus have the ability to disperse great distances. 

Image by W.P. Armstrong

In this image, you can see exactly how tiny orchid seeds tend to be. This is one capsule of an unknown orchid, carrying approximately one million seeds.

In terms of structure, Phantom Orchid seeds oddly enough do not contain a food reserve. Instead, only the embryo can be found within its shell. This would seem to account for how its seeds manage to be so small. After dispersal, the seeds need to find mycorrhizal fungi rather soon, or they will die without a food source. 

Cephalanthera austiniae is pollinated by insects, and apparently isn't very picky about which insects happen to pollinate it. The flowers give off a faint, vanilla-like scent. They have a sticky stigma, so that when insects enter the flower, the pollen becomes stuck to their bodies. The pollen is then dispersed in the next orchid that is visited. However, there has been a noticeable decline of pollinators as of recent history, most likely because of habitat destruction. Because of this, some efforts to hand-pollinate them have been made, which have resulted in successful, seed-bearing capsules.