Precise knowledge of the static density response function (SDRF) of the uniform electron gas serves as key input for numerous applications, most importantly for density functional theory beyond generalized gradient approximations. Here we extend the configuration path integral Monte Carlo (CPIMC) formalism that was previously applied to the spatially uniform electron gas to the case of an inhomogeneous electron gas by adding a spatially periodic external potential. This procedure has recently been successfully used in permutation blocking path integral Monte Carlo simulations (PB-PIMC) of the warm dense electron gas [T. Dornheim et al., Phys. Rev. E 96, 023203 (2017)], but this method is restricted to low and moderate densities. Implementing this procedure into CPIMC allows us to obtain highly accurate finite temperature results for the SDRF of the electron gas at high to moderate densities closing the gap left open by the PB-PIMC data. In this paper, we demonstrate how the CPIMC formalism can be efficiently extended to the spatially inhomogeneous electron gas and present the first data points. Finally, we discuss finite size errors involved in the quantum Monte Carlo results for the SDRF in detail and present a solution how to remove them that is based on a generalization of ground state techniques.