We consider the absorption by bound electrons of dark matter in the form of dark photons and axion-like particles, as well as of dark photons from the Sun, in current and next-generation direct detection experiments. Experiments sensitive to electron recoils can detect such particles with masses between a few eV to more than 10 keV. For dark photon dark matter, we update a previous bound based on XENON10 data and derive new bounds based on data from XENON100 and CDMSlite. We find these experiments to disfavor previously allowed parameter space. Moreover, we derive sensitivity projections for SuperCDMS at SNOLAB for silicon and germanium targets, as well as for various possible experiments with scintillating targets (cesium iodide, sodium iodide, and gallium arsenide). The projected sensitivity can probe large new regions of parameter space. For axionlike particles, the same current direction detection data improves on previously known direct-detection constraints but does not bound new parameter space beyond known stellar cooling bounds. However, projected sensitivities of the upcoming SuperCDMS SNOLAB using germanium can go beyond these and even probe parameter space consistent with possible hints from the white dwarf luminosity function. We find similar results for dark photons from the sun. For all cases, direct-detection experiments can have unprecedented sensitivity to dark-sector particles.