Pulsar wind nebulae (PWNe) are among the most extreme particle accelerators in galaxies, are recognized as multi-TeV electron/positron sources, and are one of the dominant classes of Galactic gamma-ray sources. Vela X is a nearby PWN at 290 pc from the Earth with large apparent size ($>1^\circ$). The H.E.S.S. array of imaging atmospheric Cherenkov telescopes has detected Vela~X as one of the brightest known sources of TeV gamma rays. The bulk of the gamma-ray emission measured using H.E.S.S. coincides with an elongated structure known from X-ray observations and dubbed the cocoon, that seemingly emanates from the region of the pulsar wind termination shock. The spectral energy distribution of the cocoon peaks at around 10~TeV, and then presents a cutoff that can be precisely measured with H.E.S.S. owing to the extreme brightness of the source. Electrons radiating inverse-Compton gamma rays in the cutoff region are the same responsible for the X-ray synchrotron emission at energies $> 1$ keV. Therefore, Vela X provides a unique test case, in which we can constrain the densities and spectra of accelerated leptons in the cutoff regime, as well as the magnetic field properties, with minimal modeling assumptions. Thanks to the proximity/large apparent size of the source, this can be done in a spatially-resolved fashion across the PWN. We will present an analysis of H.E.S.S. data combined with X-ray data from the $Suzaku$ space telescope. We will discuss implications for the mechanisms behind particle acceleration and transport, constrain the strength of the magnetic field in different locations in the nebula, and probe for magnetic field turbulence.