This paper extends the Vendor Managed Inventory (VMI) literature by integrating a robust optimization framework with stochastic nonlinear inverse demand, grounded in real-world industrial data. Unlike traditional VMI models that rely on deterministic linear demand assumptions and unconstrained financial planning, this study addresses volatile market environments where pricing power diminishes with quantity and demand is subject to random shocks. A stochastic joint replenishment problem is formulated in which the vendor maximizes a mean–variance profit objective under a strict global inventory budget. The model incorporates a quadratic inverse demand function with additive uncertainty, capturing stochastic consumer willingness to pay and diminishing marginal returns. Demand and cost parameters are estimated using the publicly available DataCo Smart Supply Chain dataset, enabling a transition from synthetic assumptions to data-driven nonlinear pricing analysis. A Simulation-Based Robust Particle Swarm Optimization (Sim-RPSO) algorithm is developed to solve the resulting non-convex and noisy optimization problem. Numerical experiments demonstrate that the proposed robust policy achieves a mean profit of $6,479 with a 95% Value-at-Risk of $6,333, outperforming the deterministic benchmark by $118.34 in mean profit and $129.37 in downside protection while reducing profit variability by 9.2% and inventory expenditure by 18.7%. These findings provide actionable managerial insights into balancing profitability and risk in budget-constrained, volatile supply chains.