We theoretically investigated the vector properties of quartic solitons in a pure fourth-order-dispersion birefringent fiber and a quartic-dispersion-dominant mode-locked fiber laser. We found that, compared with scalar pure quartic solitons, a vector quartic soliton (VQS) in the birefringent fiber still preserves the Gaussian shape, except for the distinctions of reduced peak power, central frequency offset, slight frequency chirp, and mitigated oscillatory tails. We also demonstrated that pulse shaping in the mode-locked laser cavity could explicitly facilitate the formation of Kelly sidebands and distortion of oscillatory tails. Furthermore, dynamical evolutions of quartic group-velocity-locked and polarization-rotating vector solitons were obtained to enrich the nonlinear community of VQSs. We believe that our elaborate findings will bring insights into both the fundamental understanding and potential applications of VQSs.