Intelligent assistants powered by Large Language Models (LLMs) can generate program and test code with high accuracy, boosting developers' and testers' productivity. However, there is a lack of studies exploring LLMs for testing Web APIs, which constitute fundamental building blocks of modern software systems and pose significant test challenges. Hence, in this article, we introduce APITestGenie, an approach and tool that leverages LLMs to generate executable API test scripts from business requirements and API specifications. In experiments with 10 real-world APIs, the tool generated valid test scripts 57% of the time. With three generation attempts per task, this success rate increased to 80%. Human intervention is recommended to validate or refine generated scripts before integration into CI/CD pipelines, positioning our tool as a productivity assistant rather than a replacement for testers. Feedback from industry specialists indicated a strong interest in adopting our tool for improving the API test process.