The coalescence of a binary neutron star pair is expected to produce gravitational waves (GW) and electromagnetic radiation, both of which may be detectable with currently available instruments. We describe a search for a predicted r-process optical transient from these mergers, dubbed the "kilonova" (KN), using griz broadband data from the Dark Energy Survey Supernova Program (DES-SN). Some models predict KNe to be redder, shorter-lived, and dimmer than supernovae (SNe), but the event rate of KNe is poorly constrained. We simulate KN and SN light curves with the Monte-Carlo simulation code SNANA to optimize selection requirements, determine search efficiency, and predict SN backgrounds. Our analysis of the first two seasons of DES-SN data results in 0 events, and is consistent with our prediction of 1.1 ± 0.2 background events based on simulations of SNe. From our prediction, there is a 33% chance of finding 0 events in the data. Assuming no underlying galaxy flux, our search sets 90% upper limits on the KN volumetric rate of 1.0 Gpc-3 yr-1 for the dimmest KN model we consider (peak i-band absolute magnitude mag) and 2.4 Gpc-3 yr-1 for the brightest ( mag). Accounting for anomalous subtraction artifacts on bright galaxies, these limits are ∼3 times higher. This analysis is the first untriggered optical KN search and informs selection requirements and strategies for future KN searches. Our upper limits on the KN rate are consistent with those measured by GW and gamma-ray burst searches.