OriStitch is a computational fabrication workflow to turn existing flat fabrics into self-folding 3D structures. Users turn fabrics into self-folding sheets by machine embroidering functional threads in specific patterns on fabrics, and then apply heat to deform the structure into a target 3D structure. OriStitch is compatible with a range of existing materials (e.g., leather, woven fabric, and denim). We present the design of specific embroidered hinges that fully close under exposure to heat. We discuss the stitch pattern design, thread and fabric selection, and heating conditions. To allow users to create 3D textiles using our hinges, we create a tool to convert 3D meshes to 2D stitch patterns automatically, as well as an end-to-end fabrication and actuation workflow. To validate this workflow, we designed and fabricated a cap (303 hinges), a handbag (338 hinges), and a cover for an organically shaped vase (140 hinges). In technical evaluation, we found that our tool successfully converted 23/28 models (textures and volumetric objects) found in related papers. We also demonstrate the folding performance across different materials (suede leather, cork, Neoprene, and felt).