We present the design and implementation of an automated data calibration and reduction pipeline for very long baseline interferometric (VLBI) observations taken at millimeter wavelengths. These short radio wavelengths provide the best imaging resolution available from ground-based VLBI networks such as the Event Horizon Telescope (EHT) and the Global Millimeter VLBI Array (GMVA) but require specialized processing owing to the strong effects from atmospheric opacity and turbulence, as well as the heterogeneous nature of existing global arrays. The pipeline builds on a calibration suite (HOPS) originally designed for precision geodetic VLBI. To support the reduction of data for astronomical observations, we have developed an additional framework for global phase and amplitude calibration that provides output in a standard data format for astronomical imaging and analysis. The pipeline was successfully used toward the reduction of 1.3 mm observations from the EHT 2017 campaign, leading to the first image of a black hole "shadow" at the center of the radio galaxy M87. In this work, we analyze observations taken at 3.5 mm (86 GHz) by the GMVA, joined by the phased Atacama Large Millimeter/submillimeter Array in 2017 April, and demonstrate the benefits from the specialized processing of high-frequency VLBI data with respect to classical analysis techniques.