The discovery of a transient kilonova following the gravitational-wave event GW170817 highlighted the critical need for coordinated rapid and wide-field observations, inference, and follow-up across the electromagnetic spectrum. In the Southern hemisphere, the Dark Energy Camera (DECam) on the Blanco 4-m telescope is well-suited to this task, as it is able to cover wide-fields quickly while still achieving the depths required to find kilonovae like the one accompanying GW170817 to $sim$500 Mpc, the binary neutron star horizon distance for current generation of LIGO/Virgo collaboration (LVC) interferometers. Here, as part of the multi-facility followup by the Global Relay of Observatories Watching Transients Happen (GROWTH) collaboration, we describe the observations and automated data movement, data reduction, candidate discovery, and vetting pipeline of our target-of-opportunity DECam observations of S190426c, the first possible neutron star--black hole merger detected via gravitational waves. Starting 7.5hr after S190426c, over 11.28,hr of observations, we imaged an area of 525,deg$^2$ ($r$-band) and 437,deg$^2$ ($z$-band); this was 16.3% of the total original localization probability and nearly all of the probability density visible from the Southern hemisphere. The machine-learning based pipeline was optimized for fast turnaround, delivering transient candidates for human vetting within 17 minutes, on average, of shutter closure. We reported nine promising counterpart candidates 2.5 hours before the end of our observations. Our observations yielded no detection of a bona fide counterpart to $m_z = 22.5$ and $m_r = 22.9$ at the 5$sigma$ level of significance, consistent with the refined LVC positioning. We view these observations and rapid inferencing as an important real-world test for this novel end-to-end wide-field pipeline.
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