In an effort to measure the Rossiter-McLaughlin effect for the TRAPPIST-1 system, we performed high-resolution spectroscopy during transits of planets e, f, and b. The spectra were obtained with the InfraRed Doppler spectrograph on the Subaru 8.2 m telescope, and were supplemented with simultaneous photometry obtained with a 1 m telescope of the Las Cumbres Observatory Global Telescope. By analyzing the anomalous radial velocities, we found the projected stellar obliquity to be λ = 1 ± 28° under the assumption that the three planets have coplanar orbits, although we caution that the radial-velocity data show correlated noise of unknown origin. We also sought evidence for the expected deformations of the stellar absorption lines, and thereby detected the "Doppler shadow" of planet b with a false-alarm probability of 1.7%. The joint analysis of the observed residual cross-correlation map including the three transits gave λ = 19± ¹³ ₁₅. These results indicate that the the TRAPPIST-1 star is not strongly misaligned with the common orbital plane of the planets, although further observations are encouraged to verify this conclusion.