Real-world molecular design remains challenging for large language model (LLM)-based agents. It requires them to interpret design contexts, satisfy multiple constraints, identify infeasible specifications, and reason over multi-step tool outputs. Existing benchmarks do not capture this complexity, focusing instead on explicit and narrow constraints, only feasible problems, and single-path solutions. To address this gap, we propose MolDesignBench, a scenario-grounded benchmark that more closely reflects real-world molecular design for evaluating tool-augmented LLM agents. MolDesignBench comprises 2K generation and optimization instances that combine implicit requirements embedded in design narratives with explicit property and functional-group constraints, including infeasible cases, and require the effective use of 17 specialized chemistry tools. Experiments across diverse frontier LLMs reveal low success rates--with the best achieving only ∼43%--and frequent failures in implicit-constraint reasoning, infeasibility detection, and tool reasoning. The corresponding fine-grained failure-mode analysis identifies implicit constraint interpretation and infeasibility detection as the primary bottlenecks, establishing MolDesignBench as a rigorous testbed to guide future research on chemical agents. The benchmark, tool interface, and evaluation code are publicly available.