Soft and bio-inspired robotics promise to imbue robots with capabilities found in the natural world.However, many of these biological capabilities are yet to be realized. For example, current vine- and root-inspired everting robots rely on centralized control outside of the robotic tendril to process sensor information and command actuation. In contrast, roots in nature control growth direction in a distributed manner, with all control, sensing, and actuation local. Such distributed control is useful for robustness and parallelization particularly while the plants search for resources (light, water, favorable soil, etc.). Here we present an approach for exploiting these biological behaviors via a thermotropic vine-inspired robot; the device uses local, material-level sensing, actuation, and control embedded in its skin to grow toward a source of heat. We present basic modeling of the concept, design details, and experimental results showing its behavior in varied heat fields. Our simple device advances vine-inspired everting robots by offering a new, distributed method of shape control, and could lead to eventual applications such as highly parallelized robots for fire-fighting or search-and-rescue operations.