In this series of papers we present an emulator-based halo model for the non-linear clustering of galaxies in modified gravity cosmologies. In the first paper, we present emulators for the following halo properties: the halo mass function, concentration-mass relation and halo-matter cross-correlation function. The emulators are trained on data extracted from the \textsc{FORGE} and \textsc{BRIDGE} suites of $N$-body simulations, respectively for two modified gravity (MG) theories: $f(R)$ gravity and the DGP model, varying three standard cosmological parameters $\Omega_{\mathrm{m0}}, H_0, \sigma_8$, and one MG parameter, either $\bar{f}_{R0}$ or $r_{\mathrm{c}}$. Our halo property emulators achieve an accuracy of $\lesssim 1\%$ on independent test data sets. We demonstrate that the emulators can be combined with a galaxy-halo connection prescription to accurately predict the galaxy-galaxy and galaxy-matter correlation functions using the halo model framework.