Vibrational overtones in deeply bound molecules are sensitive probes for variation of the proton-to-electron mass ratio μ. In nonpolar molecules, these overtones may be driven as two-photon transitions. Here, we present procedures for experiments with O₂⁺, including state-preparation through photoionization, a two-photon probe, and detection. We calculate transition dipole moments between all X²П_g vibrational levels and those of the A²П_u excited electronic state. Using these dipole moments, we calculate two-photon transition rates and AC-Stark-shift systematics for the overtones. We estimate other systematic effects and statistical precision. Two-photon vibrational transitions in O₂⁺ provide multiple routes to improved searches for μ variation.