Heterogeneous robot teams operating in realistic settings often must accomplish complex missions requiring collaboration and adaptation to information acquired online. Because robot teams frequently operate in unstructured environments -- uncertain, open-world settings without prior maps -- subtasks must be grounded in robot capabilities and the physical world. While heterogeneous teams have typically been designed for fixed specifications, generative intelligence opens the possibility of teams that can accomplish a wide range of missions described in natural language. However, current large language model (LLM)-enabled teaming methods typically assume well-structured and known environments, limiting deployment in unstructured environments. We present SPINE-HT, a framework that addresses these limitations by grounding the reasoning abilities of LLMs in the context of a heterogeneous robot team through a three-stage process. Given language specifications describing mission goals and team capabilities, an LLM generates grounded subtasks which are validated for feasibility. Subtasks are then assigned to robots based on capabilities such as traversability or perception and refined given feedback collected during online operation. In simulation experiments with closed-loop perception and control, our framework achieves nearly twice the success rate compared to prior LLM-enabled heterogeneous teaming approaches. In real-world experiments with a Clearpath Jackal, a Clearpath Husky, a Boston Dynamics Spot, and a high-altitude UAV, our method achieves an 87\% success rate in missions requiring reasoning about robot capabilities and refining subtasks with online feedback. More information is provided at https://zacravichandran.github.io/SPINE-HT.