This paper investigates a high-altitude platform (HAP) network enhanced with reconfigurable intelligent surfaces (RISs). The arbitrary placement of HAPs and RISs is modeled using stochastic geometry, specifically as homogeneous Poisson point processes. The HAP--RIS links are assumed to follow Rician fading, while the RIS--user links experience shadowed-Rician fading. The system's coverage probability and ergodic capacity are derived analytically and validated through Monte Carlo simulations. The results highlight significant performance gains and demonstrate the influence of various system parameters and fading conditions. The proposed system has potential for enhancing connectivity and data offloading in practical scenarios.