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 arcmin(2)) and the only at 2 mm to search for dusty star-forming galaxies (DSFGs). We use the 13 sources detected above 5 sigma 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 alpha(LF) = -0.42(-0.04)(+0.02). We conclude that the dust-obscured component, which peaks at z approximate to 2-2.5, has dominated the cosmic history of star formation for the past similar to 12 billion years, back to z similar to 4. At z = 5, the dust-obscured star formation is estimated to be similar to 35% of the total star formation rate density and decreases to 25%-20% at z = 6-7, implying a minor contribution of dusten-shrouded 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.