Multiscale methods are often utilized to investigate the effect of microstructure on effective macroscopic properties. Simulating practical problems with such microstructures in one scale is in general intractable, as very fine meshes would be required to sufficiently capture the geometry of the microstructure. In two-scale simulations, a separate microscopic problem is defined on a representative volume element and solved to find the average behavior of the microstructure, while the macrostructure can assume a simplified geometry with a coarse mesh. By solving the microscopic boundary value problem at every macroscopic integration point, a connection between the macro- and microscale is established. Due to the repeated solution of the microscopic problem, two-scale simulations are computationally demanding, and Reduced Order Models (ROMs), that can accurately approximate the microstructure for a wide range of parameters, are necessary. In this work, we employ a combination of the Reduced Basis Method [1] and the Empirical Cubature Method (ECM) [2] to build a ROM for the microscopic simulation. Furthermore, our ROM can treat geometrical parameterizations of the microstructure, thus allowing its application in two-scale shape optimization problems. Two numerical examples involving geometrically parameterized composite and porous microstructures, consisting of history-dependent elasto-plastic materials, are considered to validate the framework. High accuracies as well as significant speed ups are observed. [1] Quarteroni, A., Manzoni, A., Negri, F. (2015). Reduced basis methods for partial differential equations: an introduction (Vol. 92). Springer. [2] Hernandez, J. A., Caicedo, M. A., Ferrer, A. (2017). Dimensional hyper-reduction of nonlinear finite element models via empirical cubature. Computer methods in applied mechanics and engineering, 313, 687-722.