Motivated by lunar exploration, we consider deploying a network of mobile robots to explore an unknown environment while acting as a cooperative positioning system. Robots measure and communicate position-related data in order to perform localization in the absence of infrastructure-based solutions (e.g. stationary beacons or GPS). We present Trilateration for Exploration and Mapping (TEAM), a novel algorithm for low-complexity localization and mapping with robotic networks. TEAM is designed to leverage the capability of commercially-available ultra-wideband (UWB) radios on board the robots to provide range estimates with centimeter accuracy and perform anchorless localization in a shared, stationary frame. It is well-suited for feature-deprived environments, where feature-based localization approaches suffer. We provide experimental results in varied Gazebo simulation environments as well as on a testbed of Turtlebot3 Burgers with Pozyx UWB radios. We compare TEAM to the popular Rao-Blackwellized Particle Filter for Simultaneous Localization and Mapping (SLAM). We demonstrate that TEAM requires an order of magnitude less computational complexity and reduces the necessary sample rate of LiDAR measurements by an order of magnitude. These advantages do not require sacrificing performance, as TEAM reduces the maximum localization error by 50% and achieves up to a 28% increase in map accuracy in feature-deprived environments and comparable map accuracy in other settings.