Exoplanets with cloud-free, haze-free atmospheres at the pressures probed by transmission spectroscopy represent a valuable opportunity for detailed atmospheric characterization and precise chemical abundance constraints. We present the first optical to infrared (0.3-5 mu m) transmission spectrum of the hot Jupiter WASP-62b, measured with Hubble/STIS and Spitzer/IRAC. The spectrum is characterized by a 5.1 sigma detection of Na i absorption at 0.59 mu m, in which the pressure-broadened wings of the Na D-lines are observed from space for the first time. A spectral feature at 0.4 mu m is tentatively attributed to SiH at 2.1 sigma confidence. Our retrieval analyses are consistent with a cloud-free atmosphere without significant contamination from stellar heterogeneities. We simulate James Webb Space Telescope (JWST) observations, for a combination of instrument modes, to assess the atmospheric characterization potential of WASP-62b. We demonstrate that JWST can conclusively detect Na, H2O, FeH, NH3, CO, CO2, CH4, and SiH within the scope of its Early Release Science (ERS) program. As the only transiting giant planet currently known in the JWST Continuous Viewing Zone, WASP-62b could prove a benchmark giant exoplanet for detailed atmospheric characterization in the James Webb era.