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At least half of a protostar's mass is accreted in the Class 0 phase, when the central protostar is deeply embedded in a dense, infalling envelope. We present the first systematic search for outbursts from Class 0 protostars in the Orion clouds. Using photometry from Spitzer/IRAC spanning 2004 to 2017, we detect three outbursts from Class 0 protostars with $\ge 2$ mag changes at 3.6 or 4.5 $\mu$m. This is comparable to the magnitude change of a known protostellar FU Ori outburst. Two are newly detected bursts from the protostars HOPS 12 and 124. The number of detections implies that Class 0 protostars burst every 438 yr, with a 95% confidence interval of 161 to 1884 yr. Combining Spitzer and WISE/NEOWISE data spanning 2004-2019, we show that the bursts persist for more than nine years with significant variability during each burst. Finally, we use $19-100$ $\mu$m photometry from SOFIA, Spitzer and Herschel to measure the amplitudes of the bursts. Based on the burst interval, a duration of 15 yr, and the range of observed amplitudes, 3-100% of the mass accretion during the Class 0 phase occurs during bursts. In total, we show that bursts from Class 0 protostars are as frequent, or even more frequent, than those from more evolved protostars. This is consistent with bursts being driven by instabilities in disks triggered by rapid mass infall. Furthermore, we find that bursts may be a significant, if not dominant, mode of mass accretion during the Class 0 phase.