Context. Recent molecular surveys have revealed the rich gas organization of sonic-like filaments at small scales (so-called fibers) in all types of environments prior to the formation of low- and high-mass stars. These fibers form at the end of the turbulent cascade and are identified as the fine substructure within the hierarchical nature of the gas in the interstellar medium (ISM). Aims. Isolated fibers provide the subsonic conditions for the formation of low-mass stars. This paper introduces the Emergence of high-mass stars in complex fiber networks (EMERGE) project, which investigates whether complex fiber arrangements (networks) can also explain the origin of high-mass stars and clusters. Methods. We analyzed the EMERGE Early ALMA Survey including seven star-forming regions in Orion (OMC-1,2,3, and 4 South, LDN 1641N, NGC 2023, and the Flame Nebula) that were homogeneously surveyed in three molecular lines (N2H+ J = 1–0, HNC J = 1–0, and HC3N J = 10–9) and in the 3 mm continuum using a combination of interferometric ALMA mosaics and IRAM-30 m single-dish (SD) maps, together with a series of Herschel, Spitzer, and WISE archival data. We also developed a systematic data reduction framework allowing the massive data processing of ALMA observations. Results. We obtained independent continuum maps and spectral cubes for all our targets and molecular lines at different (SD and interferometric) resolutions, and we explored multiple data combination techniques. Based on our low-resolution (SD) observations (30″ or ~12 000 au), we describe the global properties of our sample, which covers a wide range of physical conditions, including low-(OMC-4 South and NGC 2023), intermediate (OMC-2, OMC-3, and LDN 1641N), and high-mass (OMC-1 and Flame Nebula) star-forming regions in different evolutionary stages. The comparison between our single-dish maps and ancillary YSO catalogs denotes N2H+ (1–0) as the best proxy for the dense, star-forming gas in our targets, which show a constant star formation efficiency and a fast time evolution of ≲1 Myr. While apparently clumpy and filamentary in our SD data, all targets show a much more complex fibrous substructure at the enhanced resolution of our combined ALMA+IRAM-30 m maps (4″.5 or ~2000 au). A large number of filamentary features at subparsec scales are clearly recognized in the high-density gas (≳ 105 cm−3) that is traced by N2H+ (1–0) directly connected to the formation of individual protostars. Surprisingly, this complex gas organization appears to extend farther into the more diffuse gas (~103−104 cm−3) traced by HNC (1–0). Conclusions. This paper presents the EMERGE Early ALMA Survey, which includes a first data release of continuum maps and spectral products for this project that are to be analysed in future papers of this series. A first look at these results illustrates the need of advanced data combination techniques between high-resolution interferometric (ALMA) and high-sensitivity, low-resolution single-dish (IRAM-30 m) datasets to investigate the intrinsic multiscale, gas structure of the ISM.