Autonomous safe navigation in unstructured and novel environments poses significant challenges, especially when environment information can only be provided through low-cost vision sensors. Although safe reactive approaches have been proposed to ensure robot safety in complex environments, many base their theory off the assumption that the robot has prior knowledge on obstacle locations and geometries. In this paper, we present a real-time, vision-based framework that constructs continuous, first-order differentiable Signed Distance Fields (SDFs) of unknown environments entirely online, without any pre-training, and is fully compatible with established SDF-based reactive controllers. To achieve robust performance under practical sensing conditions, our approach explicitly accounts for noise in affordable RGB-D cameras, refining the neural SDF representation online for smoother geometry and stable gradient estimates. We validate the proposed method in simulation and real-world experiments using a Fetch robot. Videos and supplementary material are available at https://satyajeetburla.github.io/rnbf/.