Iron oxide nanorings have great promise for biomedical applications because of their magnetic vortex state, which endows them with a low remanent magnetization while retaining a large saturation magnetization. Here we use micromagnetic simulations to predict the exact shapes that can sustain magnetic vortices, using a toroidal model geometry with variable diameter, ring thickness, and ring eccentricity. Our model phase diagram is then compared with simulations of experimental geometries obtained by electron tomography. High axial eccentricity and low ring thickness are found to be key factors for forming vortex states and avoiding net-magnetized metastable states. We also find that while defects from a perfect toroidal geometry increase the stray field associated with the vortex state, they can also make the vortex state more energetically accessible. These results constitute an important step toward optimizing the magnetic behavior of toroidal iron oxide nanoparticles.