Background: Low-energy transfer reactions in which a proton is stripped from a deuteron projectile and dropped into a target play a crucial role in the formation of nuclei in both primordial and stellar nucleosynthesis, as well as in the study of exotic nuclei using radioactive beam facilities and inverse kinematics. Ab initio approaches have been successfully applied to describe the $^3$H$(d,n)^4$He and $^3$He$(d,p)^4$He fusion processes. Purpose: An ab initio treatment of transfer reactions would also be desirable for heavier targets. In this work, we extend the ab initio description of $(d,p)$ reactions to processes with light $p$-shell nuclei. As a first application, we study the elastic scattering of deuterium on $^7$Li and the ${}^{7}$Li($d$,$p$)${}^{8}$Li transfer reaction based on a two-body Hamiltonian. Methods: We use the no-core shell model to compute the wave functions of the nuclei involved in the reaction, and describe the dynamics between targets and projectiles with the help of microscopic-cluster states in the spirit of the resonating group method. Results: The shape of the excitation functions for deuteron impinging on ${}^{7}$Li are qualitatively reproduced up to the deuteron breakup energy. The interplay between $d$-$^7$Li and $p$-$^8$Li particle-decay channels determines some features of the ${}^{9}$Be spectrum above the $d$+${}^{7}$Li threshold. Our prediction for the parity of the 17.298 MeV resonance is at odds with the experimental assignment Conclusions: Deuteron stripping reactions with $p$-shell targets can now be computed ab initio, but calculations are very demanding. A quantitative description of the ${}^{7}$Li($d$,$p$)${}^{8}$Li reaction will require further work to include the effect of three-nucleon forces and additional decay channels, and improve the convergence rate of our calculations.