Photoionization cross sections and rate coefficients have been calculated for all bound vibrational levels of the 1sσ
g state of
H
2
+
, HD+, and
D
2
+
. The Born–Oppenheimer approximation is employed in our calculation of vibrationally resolved photoionization cross sections. Vibrationally resolved and local thermal equilibrium photoionization rate coefficients are presented for photon temperatures less than 50,000 K and are found to be several orders of magnitude larger than previous results in the literature. Analytic fits for the vibrationally resolved and local thermal equilibrium photoionization rate coefficients are provided. Near-threshold oscillations in the vibrationally resolved photoionization are observed. A benchmark set of photoionization cross sections are presented. Fixed-nuclei photoionization cross sections are calculated using two-center true continuum wave functions and are verified by comparison with previous calculations and are found to be in excellent agreement in all cases. Data files for our set of benchmark cross sections, rate coefficients, and fitting parameters for
H
2
+
, HD+, and
D
2
+
are available on Zenodo under an open-source Creative Commons Attribution license at doi:10.5281/zenodo.8304061.