In this series of two papers, we describe a project with the Space Telescope Imaging Spectrograph (STIS) on the Hubble Space Telescope (HST) to measure the line-of-sight velocities of stars in the central few arcseconds of the dense globular cluster M15. This first paper focuses on the observations and reduction of the data. We "scanned" the central region of M15 spectroscopically by consecutively placing the 0.1" HST/STIS slit at 18 adjacent positions. The spectral pixel size exceeds the velocity dispersion of M15. This puts the project at the limit of what is feasible with STIS, and exceedingly careful and complicated data reduction and analysis were required.