In this paper, a sampling-based Stochastic Model Predictive Control algorithm is proposed for discrete-time linear systems subject to both parametric uncertainties and additive disturbances. One of the main drivers for the development of the proposed control strategy is the need of real-time implementability of guidance and control strategies for automated rendezvous and proximity operations between spacecraft. The paper presents considers the validation of the proposed control algorithm on an experimental testbed, showing how it may indeed be implemented in a realistic framework. Parametric uncertainties due to the mass variations during operations, linearization errors, and disturbances due to external space environment are simultaneously considered. The approach enables to suitably tighten the constraints to guarantee robust recursive feasibility when bounds on the uncertain variables are provided, and under mild assumptions, asymptotic stability in probability of the origin can be established. The offline sampling approach in the control design phase is shown to reduce the computational cost, which usually constitutes the main limit for the adoption of Stochastic Model Predictive Control schemes, especially for low-cost on-board hardware. These characteristics are demonstrated both through simulations and by means of experimental results.