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A dark photon is one of the simplest extensions of the Standard Model of particle physics and can be a dark matter candidate. Dark photons kinetically mix with ordinary photons. The mass range from $10^{-4}$ to $10^{-3}$ eV of such dark photons is under-constrained by laboratory-based experiments and a new search is therefore motivated. In this mass range, dark photons behave like waves rather than particles and the corresponding electromagnetic waves are in the millimeter-wave range. The technical difficulties of the millimeter waves have prevented so far dark photon experiments in this mass range. We propose the use of coherent millimeter waves to search for dark photons in a Light-Shining-through-a-Wall (LSW) experiment. We clarify the merit and limitations of coherent wave detection and briefly investigate the potential of single photon sensors at microwaves. Development of millimeter-wave technology is not only limited to dark photons. Technically, an experiment for dark photons by using electromagnetic waves resembles that for axions, another light dark matter candidate, with static magnetic fields. This paper represents an essential step towards axion LSW in the millimeter-wave range (STAX experiment) as a potential successor of an on-going experiment in infrared.