Park-and-ride systems have the potential to improve the efficiency of transportation networks by providing targeted shared mobility services. The design of a park-and-ride system depends on its role in regards to the broader transportation network. Reconfigurable park-and-ride systems aim to provide complementary shared mobility services in the context of varying travel demand scenarios, such as special events, network maintenance operations or non-recurrent perturbations. The design of reconfigurable park-and-ride systems involves the location of access and egress hubs for shared mobility services. We study an extended version of this hub location problem with integrated fleet assignment decisions. We consider stochastic scenarios representative of varying travel demand and traffic conditions on the network and propose a two-stage stochastic integer programming hub location formulation for this problem. First-stage variables represent hub location decision while second-stage variables represent both scenario-based transportation flows and fleet assignment decisions. The latter represent shared mobility service vehicles and they are modeled as integer variables. We develop solution methods to solve this two-stage stochastic integer programming hub location formulation. Exact approaches based on the L-shaped method are proposed with single- and multi-cut configurations. Valid inequalities along with a tight lower bound for the generation of optimality cuts are presented. We also develop a matheuristic to solve larger problem instances. We report numerical results on problem instances based on real data of the city of Lyon, France. We show how stochastic scenarios representative of varying demand and traffic conditions can be generated from such data. Our experiments demonstrate the benefits of this integrated modeling approach for designing efficient reconfigurable park-and-ride systems while considering fleet assignment decisions.