Computational modeling of cardiovascular function has become a critical part of diagnosing, treating and understanding cardiovascular disease. Most strategies involve constructing anatomically accurate computer models of cardiovascular structures, which is a multistep, time-consuming process. To improve the model generation process, we herein present SeqSeg (sequential segmentation): a novel deep learning-based automatic tracing and segmentation algorithm for constructing image-based vascular models. SeqSeg leverages local U-Net-based inference to sequentially segment vascular structures from medical image volumes. We tested SeqSeg on CT and MR images of aortic and aortofemoral models and compared the predictions to those of benchmark 2D and 3D global nnU-Net models, which have previously shown excellent accuracy for medical image segmentation. We demonstrate that SeqSeg is able to segment more complete vasculature and is able to generalize to vascular structures not annotated in the training data.