The discovery of a transient kilonova following the gravitational-wave (GW) 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 ̃500 Mpc, the binary neutron star (NS) horizon distance for current generation of LIGO/Virgo collaboration (LVC) interferometers. Here, as part of the multi-facility follow-up by the Global Relay of Observatories Watching Transients Happen 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 NS-black hole merger detected in GWs. Starting 7.5 hr 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 visible from the southern hemisphere. The machine-learning-based pipeline was optimized for fast turnaround, delivering transients for human vetting within 17 minutes, on average, of shutter closure. We reported nine promising counterpart candidates 2.5 hr before the end of our observations. One hour after our data-taking ended (roughly 20 hr after the announcement of S190426c), LVC released a refined skymap that reduced the probability coverage of our observations to 8.0%, demonstrating a critical need for localization updates on shorter (̃hour) timescales. Our observations yielded no detection of a bona fide counterpart to m $_ z $ = 21.7 and m $_ r $ = 22.2 at the 5σ 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.