Cover crops (CCs) are regarded as beneficial to agricultural practice as an option for soil quality improvement in field production systems. The main goal of this study was to assess the impact of spring phacelia (Phacelia tanacetifolia Benth.) and buckwheat (Fagopyrum Mill.) in a crop rotation (CC–leek–parsley, 2020–2021) on the physicochemical and biological properties of the soil in an organic high tunnel system. Soil analyses involved measurements of bulk density, water capacity, soil aggregation, soil organic carbon (SOC), available soil nutrients, as well as microbial abundance and diversity. Phacelia generated more aboveground biomass (58.2 t fresh matter ha−1) than buckwheat (33.0 t ha−1), and their biomass contained 161 kg N ha−1 and 67 kg N ha−1, respectively. A large quantity of elements, such as N, Ca, P, S, B, and Cu, were found in phacelia biomass. More Mg and Na were found in buckwheat plants. The results showed that CC biomass significantly improved some of the soil physical and chemical properties, such as soil organic carbon stock and wet aggregate stability, and decreased soil bulk density. Cover crop treatments changed the dynamics of soil bacterial and fungus populations in a high tunnel system. Phacelia increased the quantity of ammonifiers and nitrifiers in the soil substantially. Further research with a long-term focus is needed to assess the impact of cover crops on soil properties, soil quality, and subsequent crop yields in high tunnel crop rotation and management systems.