In this paper we present a neurosymbolic architecture for coupling language-guided visual reasoning with robot manipulation. A non-expert human user can prompt the robot using unconstrained natural language, providing a referring expression (REF), a question (VQA), or a grasp action instruction. The system tackles all cases in a task-agnostic fashion through the utilization of a shared library of primitive skills. Each primitive handles an independent sub-task, such as reasoning about visual attributes, spatial relation comprehension, logic and enumeration, as well as arm control. A language parser maps the input query to an executable program composed of such primitives, depending on the context. While some primitives are purely symbolic operations (e.g. counting), others are trainable neural functions (e.g. visual grounding), therefore marrying the interpretability and systematic generalization benefits of discrete symbolic approaches with the scalability and representational power of deep networks. We generate a 3D vision-and-language synthetic dataset of tabletop scenes in a simulation environment to train our approach and perform extensive evaluations in both synthetic and real-world scenes. Results showcase the benefits of our approach in terms of accuracy, sample-efficiency, and robustness to the user's vocabulary, while being transferable to real-world scenes with few-shot visual fine-tuning. Finally, we integrate our method with a robot framework and demonstrate how it can serve as an interpretable solution for an interactive object-picking task, achieving an average success rate of 80.2\%, both in simulation and with a real robot. We make supplementary material available in https://gtziafas.github.io/neurosymbolic-manipulation.