I asked Dennis Whigham (Smithsonian Environmental Research Center / North American Orchid Conservation Center) to offer a response to your question, so here is what he had to say:
Seed storage requirements have been developed for very few native orchids. The most recent compilation of relevant information is in Swarts and Dixon, which is available on Amazon (Swarts, N. D., & Dixon, K. W. (2017). Conservation methods for terrestrial orchids). There is some helpful information on seed collection and storage in the book.
One suggestion would be to put seeds (if you have enough) in a few different conditions. First you need to reduce the moisture content of the seeds before storage. This can be done by drying them in a room with controlled humidity (15% relative humidity) or in containers that contain material that will reduce the moisture content of the seeds to the appropriate level (e.g., saturated solution of lithium chloride or over dried rice). Seeds should be put into these conditions for about 10 days. They can then be stored in a the frig at 4C, in a -20C freezer or, if possible, in a facility where they can stored on liquid nitrogen. Assuming that the seeds will be in small containers that may be in larger containers, silica gel with a color indicator needs to be put into the containers in which the seeds (not directly with the seeds) are stored to make sure that the moisture content does not change.
Until detailed requirements are worked out for Piper (Platanthera) michelli, do as much as you can with the seeds. Once they are stored, you can bring some out at different times (6 months, a year, ….) and test them for viability or start to germinate them. Hopefully, you project is also resulting in the collection of roots from which orchid mycorrhiza can be isolated, grown and cultured for used in future germination studies or growth/restoration experiments.