We present the first results from the Mapping Obscuration to Reionization with ALMA (MORA) survey, the largest Atacama Large Millimeter/submillimeter Array (ALMA) blank-field contiguous survey to date (184 arcmin2) and the only at 2 mm to search for dusty star-forming galaxies (DSFGs). We use the 13 sources detected above 5σ to estimate the first ALMA galaxy number counts at this wavelength. These number counts are then combined with the state-of-the-art galaxy number counts at 1.2 and 3 mm and with a backward evolution model to place constraints on the evolution of the IR luminosity function and dust-obscured star formation in the past 13 billion years. Our results suggest a steep redshift evolution on the space density of DSFGs and confirm the flattening of the IR luminosity function at faint luminosities, with a slope of We conclude that the dust-obscured component, which peaks at z ≈ 2-2.5, has dominated the cosmic history of star formation for the past ∼12 billion years, back to z ∼ 4. At z = 5, the dust-obscured star formation is estimated to be ∼35% of the total star formation rate density and decreases to 25%-20% at z = 6-7, implying a minor contribution of dust-enshrouded star formation in the first billion years of the universe. With the dust-obscured star formation history constrained up to the end of the epoch of reionization, our results provide a benchmark to test galaxy formation models, to study the galaxy mass assembly history, and to understand the dust and metal enrichment of the universe at early times.