The importance of sustainability in supply chain management is growing worldwide. It is possible to find reasons for this using various phenomena that negatively affect humanity, e.g., climate change, scarce materials, supply disruptions, and complex fossil fuel dependency. Because of that, is extremely important to constantly look for new ways to systematically increase sustainability in enterprises and their logistics and supply chain processes by considering different stakeholders and influential factors. Therefore, this paper explores how different types of organizational culture and normative commitment impact sustainability and each other in business logistics and supply chains and develops a conceptual model to manage this challenge. Gaining new insights is valuable especially for managers to obtain better information on how to improve sustainability not just by integrating green technologies but mainly by changing culture, attitude, and perception in their enterprises. The research is focused on employees from global logistics or related branches in micro, small, medium, and large enterprises with the primary activity mostly related to manufacturing, transport, and storage. The findings are based on the questionnaire which was sent directly to 1576 employees from 528 enterprises. A total of 516 employees from enterprises that are mostly located in 34 countries responded to requests for participation. The results reveal statistically significant positive and negative impacts, e.g., clan culture has a positive statistically significant impact on the sustainable development of supply chains. Most of the connections to the eighth Sustainable Development Goal by the United Nations (decent work and economic growth) were also found, which was the enterprise’s highest priority with a share of 52.99%. A contribution to the theory development is gained using the developed model that considers both positive and negative statistically significant impacts studied.