Spin-orbital-angular-momentum (SOAM) coupling has been realized in recent experiments of Bose-Einstein condensates [Chen et al., Phys. Rev. Lett. 121, 113204 (2018) and Zhang et al., Phys. Rev. Lett. 122, 110402 (2019)], where the orbital angular momentum imprinted upon bosons directly leads to quantized vortices. However, an s-wave Fermi pairing superfluid under the same SOAM coupling is typically vortex-less, as the two fermion species acquire opposite angular momenta in the center-of-mass motion. Here we show that, by introducing a moderate two-photon detuning in the Raman process generating the SOAM coupling, a quantized vortex, with size comparable to the beam waist of Raman lasers, can be stabilized in a Fermi superfluid. Such a giant vortex state can be viewed as the angular analogue of the Fulde-Ferrell states under a spin-orbit-coupling-induced Fermi-surface deformation, now with Cooper pairs carrying quantized angular momenta. Due to the spin-polarized nature of vortex bound states, these vortices feature a large spin polarization at the vortex core, thus providing an ideal signal for their experimental detection.