The Sloan Digital Sky Survey V (SDSS-V) is an all-sky spectroscopic survey of <6 million objects, designed to decode the history of the Milky Way, reveal the inner workings of stars, investigate the origin of solar systems, and track the growth of supermassive black holes across the Universe. The Local Volume Mapper (LVM) is one of three surveys that form SDSS-V. LVM will employ a coordinated system of four telescopes feeding three fiber spectrographs at Las Campanas Observatory in Chile. The goal is to map approximately 2500 square degrees of the Galactic plane over the wavelength range 360-980 nm with R~4000 spectral resolution. These observations will reveal for the first time how distinct gaseous environments within the Galaxy interact with each other and with the stellar population, producing the large-scale interstellar medium that we observe. Accurately mapping and calibrating a substantial portion of the sky at this spatial resolution requires a unique type of telescope system. Each of the four LVM telescopes has a diameter of 16 cm, making them considerably smaller and lighter than the instruments they feed. One telescope will host the science IFU containing ~1800 fibers arranged in a close-packed hexagon. Two additional Calibration telescopes will observe fields adjacent to the science IFU, in order to calibrate out terrestrial airglow and other geo-coronal emission. The fourth, Spectrophotometric telescope will make rapid observations of bright stars (typically 12 during a single IFU / Calibration exposure) to correct for telluric absorption lines and overall extinction. The fibers from all three types of telescope will be interspersed in the entrance slits of the spectrographs, allowing for simultaneous science and calibration exposures. Although considerably smaller than the next generation of giants, the LVM telescopes must also operate close to the limits of physical optics, and the geometry and scope of the LVM survey present unique challenges. For example, with this type of telescope at the Las Campanas site, the effects of optical aberrations, diffraction, seeing, and (uncorrected) atmospheric dispersion are all of comparable scale. This, coupled with the need for repeated and reliable measurements over years, leads to some unconventional design choices. This paper presents the preliminary design of the LVM telescope system and discusses the requirements and tradeoffs that led to the baseline choices.