We present experimental results of vacuum laser acceleration (VLA) of electrons using radially polarized laser pulses interacting with a plasma mirror. Tightly focused, radially polarized laser pulses have been proposed for electron acceleration because of their strong longitudinal electric field, making them ideal for VLA. However, experimental results have been limited until now because injecting electrons into the laser field has remained a considerable challenge. Here, we demonstrate experimentally that using a plasma mirror as an injector solves this problem and permits us to inject electrons at the ideal phase of the laser, resulting in the acceleration of electrons along the laser propagation direction while reducing the electron beam divergence compared to the linear polarization case. We obtain electron bunches with few-MeV energies and a 200-pC charge, thus demonstrating, for the first time, electron acceleration to relativistic energies using a radially polarized laser. High-harmonic generation from the plasma surface is also measured, and it provides additional insight into the injection of electrons into the laser field upon its reflection on the plasma mirror. Detailed comparisons between experimental results and full 3D simulations unravel the complex physics of electron injection and acceleration in this new regime: We find that electrons are injected into the radially polarized pulse in the form of two spatially separated bunches emitted from the p-polarized regions of the focus. Finally, we leverage on the insight brought by this study to propose and validate a more optimal experimental configuration that can lead to extremely peaked electron angular distributions and higher energy beams.