I suppose that by the "spacetime 4-vector", you mean $x^\mu$, a location of an event in spacetime, and your question is pretty much why the scattering amplitudes and cross sections are being measured as functions of the 4-momenta and not 4-positions.
That's easy. All the collisions appear at fixed places in the middle of the detectors, so the spatial components of $x^\mu$ are completely determined, and the temporal coordinate $t$ is also pretty much determined by the bunching and it is irrelevant due to the time translational symmetry, anyway.
All particles that come into the collision or come out of it have almost the same position shortly before and shortly after the collision so we can't use these positions to parameterize the state of the particles. A longer time before/after the collision, the position of the particles is given by $x^\mu \sim p^\mu \cdot \Delta t_p / m_0$ so up to some mass-dependent renormalization, it is determined by the momentum (via velocities). So even if we wanted to describe the state of the initial and final particles by positions, the positions ultimately boil down to the momentum 4-vector.
The momentum 4-vector of a single particle is conserved which is why it may be uniquely associated with the particles. Quite generally, accelerators are measuring "almost directly" cross section (and related quantities) as functions of the momenta while the interpretation in the position space is "less direct" for the reasons above.