Line-intensity mapping (LIM) of emission from star-forming galaxies can be used to measure the baryon acoustic oscillation (BAO) scale as far back as the epoch of reionization. This provides a standard cosmic ruler to constrain the expansion rate of the Universe at redshifts which cannot be directly probed otherwise. In light of growing tension between measurements of the current expansion rate using the local distance ladder and those inferred from the cosmic microwave background, extending the constraints on the expansion history to bridge between the late and early Universe is of paramount importance. Using a newly derived methodology to robustly extract cosmological information from LIM, which minimizes the inherent degeneracy with unknown astrophysics, we show that present and future experiments can gradually improve the measurement precision of the expansion rate history, ultimately reaching percent-level constraints on the BAO scale. Specifically, we provide detailed forecasts for the SPHEREx satellite, which will target the H α  and Lyman- α  lines, and for the ground-based COMAP instrument---as well as a future stage-3 experiment---that will target the CO rotational lines. Besides weighing in on the so-called Hubble tension, reliable LIM cosmic rulers can enable wide-ranging tests of dark matter, dark energy and modified gravity.