This paper presents a conical conformal transmitarray (CCTA) with high transmission double C-shaped grating metasurface for generating orbital angular momentum (OAM) vortex waves. A double C-shaped grating metasurface element is proposed to achieve the 360° phase coverage in the 11-13 GHz bandwidth with a high transmission coefficient of more than 0.92. The ring layout method is adopted for the CCTA and its corresponding phase compensation formula is given. Additionally, to ensure the uniformity and consistency of coupling between adjacent elements, a calculation formula for the element number of each ring is derived. To validate the conformal design method, three prototypes of the proposed CCTA were simulated, fabricated, and measured to generate OAM vortex beams, respectively. The measured results indicate that the proposed CCTAs are in good agreement with the simulated results, verifying the effectiveness of the method. The proposed CCTAs will pave the way for an OAM-generating design method on conformal platforms for future communications and detection applications.