Quantum field theory predicts that the vacuum exhibits a nonlinear response to strong electromagnetic fields. This fundamental tenet has remained experimentally challenging and is yet to be tested in the laboratory. We present proof of concept and detailed theoretical analysis of an experimental setup for precision measurements of the quantum vacuum signal generated by the collision of a brilliant x-ray probe with a high-intensity pump laser. The signal features components polarized parallel and perpendicularly to the incident x-ray probe. Our proof-of-concept measurements show that the background can be efficiently suppressed by many orders of magnitude which should not only facilitate a detection of the perpendicularly polarized component of the nonlinear vacuum response, but even make the parallel polarized component experimentally accessible for the first time. Remarkably, the angular separation of the signal from the intense x-ray probe enables precision measurements even in the presence of pump fluctuations and alignment jitter. This provides direct access to the low-energy constants governing light-by-light scattering.