Results are presented from the first cometary observations using the Atacama Large Millimeter/Submillimeter Array (ALMA), including measurements of the spatially resolved distributions of HCN, HNC, H_2_CO, and dust within the comae of two comets: C/2012 F6 (Lemmon) and C/2012 S1 (ISON), observed at heliocentric distances of 1.5 AU and 0.54 AU, respectively. These observations (with angular resolution ~0.5"), reveal an unprecedented level of detail in the distributions of these fundamental cometary molecules, and demonstrate the power of ALMA for quantitative measurements of the distributions of molecules and dust in the inner comae of typical bright comets. In both comets, HCN is found to originate from (or within a few hundred kilometers of) the nucleus, with a spatial distribution largely consistent with spherically symmetric, uniform outflow. By contrast, the HNC distributions are clumpy and asymmetrical, with peaks at cometocentric radii ~500-1000km, consistent with release of HNC in collimated outflow(s). Compared to HCN, the H_2_ CO distribution in comet Lemmon is very extended. The interferometric visibility amplitudes are consistent with coma production of H_2_CO and HNC from unidentified precursor material(s) in both comets. Adopting a Haser model, the H_2_CO parent scale length is found to be a few thousand kilometers in Lemmon and only a few hundred kilometers in ISON, consistent with the destruction of the precursor by photolysis or thermal degradation at a rate that scales in proportion to the solar radiation flux.