Local luminosity functions are fundamental benchmarks for high-redshift galaxy formation and evolution studies as well as for models describing these processes. Determining the local luminosity function in the submillimeter range can help to better constrain in particular the bolometric luminosity density in the local Universe, and Herschel offers the first opportunity to do so in an unbiased way by imaging large sky areas at several submillimeter wavelengths. We present the first Herschel measurement of the submillimeter 0 < z < 0.2 local luminosity function and infrared bolometric (8-1000 mu m) local luminosity density based on SPIRE data from the HerMES Herschel key program over 14.7 deg(2). Flux measurements in the three SPIRE channels at 250, 350 and 500 mu m are combined with Spitzer photometry and archival data. We fit the observed optical-to-submillimeter spectral energy distribution of SPIRE sources and use the 1/V-max estimator to provide the first constraints on the monochromatic 250, 350 and 500 mu m as well as on the infrared bolometric (8-1000 mu m) local luminosity function based on Herschel data. We compare our results with modeling predictions and find a slightly more abundant local submillimeter population than predicted by a number of models. Our measurement of the infrared bolometric (8-1000 mu m) local luminosity function suggests a flat slope at low luminosity, and the inferred local luminosity density, 1.31(-0.21)(+ 0.24) x 10(8) L-circle dot Mpc(-3), is consistent with the range of values reported in recent literature.