Delivery of diffusible nutrients and drugs in tissues is limited in part by the distance over which substances must diffuse between the vascular space and the surrounding tissues and by upstream losses prior to local delivery by the blood. By examining the fractal behavior of two-dimensional vascular networks in the murine dorsal skinfold chamber preparation, we have identified distinct architectural features of normal and tumor vascular networks that lead to fundamentally different transport behavior. Normal capillaries which are relatively straight and regularly spaced are well modeled by the widely used Krogh cylinder model. In contrast, the fractal dimensions of tumor vascular networks suggest that the tortuous vessels and wide range of avascular spaces found in tumors are better represented by invasion percolation, a well-known statistical growth process governed by local substrate properties. Based on these observations, we have constructed a percolation-based model of tumor vascular growth that enables us to predict the effects of network architecture on transport. We find that the number of avascular spaces in tumors scales with the size of the spaces so that there will exist a few large avascular spaces and many smaller avascular spaces between vessels. We also find that the tortuosity of the vessels, as reflected by the elevated minimum path dimension, produces regions of locally flow-limited transport and reduces flow through the tumor as a whole. Our model helps to explain the long-standing paradox that tumor vasculature has a higher geometrical resistance than normal vasculature despite increases in vessel diameter. A comparison to oxygenation measurements in normal and tumor tissues shows that our model predicts the architectural obstacles to transport in tumors more accurately than the Krogh cylinder model. Our results suggest that clinical interventions that yield more regular vascular geometry may be useful as a supplement to those that improve arterial availability or decrease rates of consumption by the tissue.