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This work continues our program of lattice-QCD baryon physics using staggered fermions for both the sea and valence quarks. We present a proof-of-concept study that demonstrates, for the first time, how to calculate baryon matrix elements using staggered quarks for the valence sector. We show how to relate the representations of the continuum staggered flavor-taste group $\text{SU}(8)_{FT}$ to those of the discrete lattice symmetry group. The resulting calculations yield the normalization factors relating staggered baryon matrix elements to their physical counterparts. We verify this methodology by calculating the isovector vector and axial-vector charges $g_V$ and $g_A$. We use a single ensemble from the MILC Collaboration with 2+1+1 flavors of sea quark, lattice spacing $a\approx 0.12$ fm, and a pion mass $M_π\approx305$ MeV. On this ensemble, we find results consistent with expectations from current conservation and neutron beta decay. Thus, this work demonstrates how highly-improved staggered quarks can be used for precision calculations of baryon properties, and, in particular, the isovector nucleon charges.