In 1922 the Dutch astronomer Jacobus Kapteyn  studied the vertical motions of all known stars near the Galactic plane and used these data to calculate the acceleration of matter. This amounts to treating the stars as members of a "star atmosphere", a statistical ensemble in which the density of stars and their velocity dispersion defines a "temperature" from which one obtains the gravitational potential. This is analogous to how one obtains the gravitational potential of the Earth from a study of the atmosphere. Kapteyn found that the spatial density is sufficient to explain the vertical motions.
Later in the same year the British astronomer James Jeans  reanalyzed Kapteyn's data and found a mass deficit: to each bright star two dark stars had to be present. The result contradicted grossly the expectations: if the potential provided by the known stars was not sufficient to keep the stars bound to the Galactic disk, the Galaxy should rapidly be losing stars. Since the Galaxy appeared to be stable there had to be some missing matter near the Galactic plane.
In 1932 the Dutch astronomer Jan Hendrik Oort  reanalyzed the vertical motions and came to the same conclusion as Jeans. There was indeed a mass deficit which Oort proposed to indicate the presence of some dark matter in our Galaxy. The possibility that this missing matter would be nonbaryonic could not even be thought of at that time. Note that the first neutral baryon, the neutron, was discovered by James Chadwick  only in the same year, in 1932.
However, it is nowadays considered, that this does not prove the existence of DM in the disk. The potential in which the stars are moving is not only due to the disk, but rather to the totality of matter in the Galaxy which is dominated by the Galactic halo. The advent of much more precise data in 1998 led Holmberg & Flynn  to conclude that no DM was present in the disk.
Oort determined the mass of the Galaxy to be 1011 Msun, and thought that the nonluminous component was mainly gas. Still in 1969 he thought that intergalactic gas made up a large fraction of the mass of the universe . The general recognition of the missing matter as a possibly new type of non-baryonic DM dates to the early eighties.