Foundation models like Vision-Language Models (VLMs) excel at common sense vision and language tasks such as visual question answering. However, they cannot yet directly solve complex, long-horizon robot manipulation problems requiring precise continuous reasoning. Task and Motion Planning (TAMP) systems can handle long-horizon reasoning through discrete-continuous hybrid search over parameterized skills, but rely on detailed environment models and cannot interpret novel human objectives, such as arbitrary natural language goals. We propose integrating VLMs into TAMP systems by having them generate discrete and continuous language-parameterized constraints that enable open-world reasoning. Specifically, we use VLMs to generate discrete action ordering constraints that constrain TAMP search over action sequences, and continuous constraints in the form of code that augments traditional TAMP manipulation constraints. Experiments show that our approach, OWL-TAMP, outperforms baselines relying solely on TAMP or VLMs across several long-horizon manipulation tasks specified directly in natural language. We additionally demonstrate that OWL-TAMP can be deployed with an off-the-shelf TAMP system to solve challenging manipulation tasks on real-world hardware.