Context. The X-ray luminosities of T Tauri stars are about two to four orders of magnitude higher than the luminosity of the contemporary Sun. As these stars are born in clusters, their disks are not only irradiated by their parent star but also by an X-ray background field produced by the cluster members. Aims. We aim to quantify the impact of X-ray background fields produced by young embedded clusters on the chemical structure of disks. Further, we want to investigate the importance of the dust for X-ray radiative transfer in disks. Methods. We present a new X-ray radiative transfer module for the radiation thermo-chemical disk code PRODIMO (PROtoplanetary DIsk MOdel), which includes X-ray scattering and absorption by both the gas and dust component. The X-ray dust opacities can be calculated for various dust compositions and dust-size distributions. For the X-ray radiative transfer we consider irradiation by the star and by X-ray background fields. To study the impact of X-rays on the chemical structure of disks we use the well established disk ionization tracers N₂H⁺ and HCO⁺. Results. For evolved dust populations (e.g. grain growth), X-ray opacities are mostly dominated by the gas; only for photon energies E ≳ 5−10 keV do dust opacities become relevant. Consequently the local disk X-ray radiation field is only affected in dense regions close to the disk midplane. X-ray background fields can dominate the local X-ray disk ionization rate for disk radii r ≳ 20 au. However, the N₂H⁺ and HCO⁺ column densities are only significantly affected in cases of low cosmic-ray ionization rates (≲10^-19 s^-1), or if the background flux is at least a factor of ten higher than the flux level of ≈10^-5 erg cm^-2 s^-1 expected for clusters typical for the solar vicinity. Conclusions. Observable signatures of X-ray background fields in low-mass star-formation regions, like Taurus, are only expected for cluster members experiencing a strong X-ray background field (e.g. due to their location within the cluster). For the majority of the cluster members, the X-ray background field has relatively little impact on the disk chemical structure.