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 (Reynolds et al. 2023, Mori et al. 2023), 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.