HEX-P is a probe-class mission concept that will combine high spatial resolution X-ray imaging (<10″ FWHM) and broad spectral coverage (0.2–80 keV) with an effective area far superior to current facilities (including XMM-Newton and NuSTAR) to enable revolutionary new insights into a variety of important astrophysical problems. With the recent discoveries of over 40 ultra-high-energy gamma-ray sources (detected above 100 TeV) and neutrino emission in the Galactic Plane, we have entered a new era of multi-messenger astrophysics facing the exciting reality of Galactic PeVatrons. In the next decade, as more Galactic PeVatrons and TeV gamma-ray sources are expected to be discovered, the identification of their acceleration and emission mechanisms will be the most pressing issue in both particle and high-energy astrophysics. In this paper, along with its companion papers, we will present that HEX-P is uniquely suited to address important problems in various cosmic-ray accelerators, including Galactic PeVatrons, through investigating synchrotron X-ray emission of TeV–PeV electrons produced by both leptonic and hadronic processes. For Galactic PeVatron candidates and other TeV gamma-ray sources, HEX-P can fill in a large gap in the spectral-energy distributions (SEDs) of many objects observed in radio, soft X-rays, and gamma rays, constraining the maximum energies to which electrons can be accelerated, with implications for the nature of the Galactic PeVatrons and their contributions to the spectrum of Galactic cosmic rays beyond the knee at ∼3 PeV. In particular, X-ray observation with HEX-P and TeV observation with CTAO will provide the most powerful multi-messenger diagnostics to identify Galactic PeVatrons and explore a variety of astrophysical shock mechanisms. We present simulations of each class of Galactic TeV–PeV sources, demonstrating the power of both the imaging and spectral capabilities of HEX-P to advance our knowledge of Galactic cosmic-ray accelerators. In addition, we discuss HEX-P’s unique and complementary roles to upcoming gamma-ray and neutrino observatories in the 2030s.