Transplantation of in-vitro-generated organ buds is a promising approach toward regenerating functional and vascularized organs. Though it has been recently shown in the context of liver models, demonstrating the applicability of this approach to other systems by delineating the molecular mechanisms guiding organ bud formation is critical. Here, we demonstrate a generalized method for organ bud formation from diverse tissues by combining pluripotent stem cell-derived tissue-specific progenitors or relevant tissue samples with endothelial cells and mesenchymal stem cells (MSCs). The MSCs initiated condensation within these heterotypic cell mixtures, which was dependent upon substrate matrix stiffness. Defining optimal mechanical properties promoted formation of 3D, transplantable organ buds from tissues including kidney, pancreas, intestine, heart, lung, and brain. Transplanted pancreatic and renal buds were rapidly vascularized and self-organized into functional, tissue-specific structures. These findings provide a general platform for harnessing mechanical properties to generate vascularized, complex organ buds with broad applications for regenerative medicine.