This paper presents the Local Volume Mapper (LVM) telescope system. LVM is one of three surveys that form the fifth generation of the Sloan Digital Sky Survey (SDSS-V), and it employs a coordinated network of four, 16 cm diameter telescopes feeding integral field units (IFUs) leading to three fiber spectrographs. One telescope hosts the science IFU, while two others observe adjacent fields to calibrate geocoronal emission. The fourth telescope makes rapid observations of bright stars to compensate for telluric absorption. The entrance slits of the spectrographs intersperse the fibers from all three types of telescope, producing truly simultaneous science and calibration exposures. We describe the final design of the telescope system and report on its construction, alignment and testing in the laboratory, as well as the integration, testing, commissioning, and actual on-sky performance at Las Campanas Observatory.