  The chiral QED$_3$--Gross-Neveu-Yukawa (QED$_3$-GNY) theory is a
$2+1$-dimensional U(1) gauge theory with $N_f$ flavors of four-component Dirac
fermions coupled to a scalar field. For $N_f=1$, the specific chiral Ising
QED$_3$-GNY model has recently been conjectured to be dual to the deconfined
quantum critical point that describes Neel--valence-bond-solid transition of
frustrated quantum magnets on square lattice. We study the universal critical
behaviors of the chiral QED$_3$-GNY model in $d=4-\epsilon$ dimensions for an
arbitrary $N_f$ . We calculate the boson anomalous dimensions, inverse
correlation length exponent, as well as the scaling dimensions of nonsinglet
fermion bilinear in the chiral QED$_3$-GNY model. The Pad$\acute{e}$ estimates
for the exponents are obtained in the chiral Ising-, XY- and
Heisenberg-QED$_3$-GNY universality class respectively. We also establish the
general condition of the supersymmetric criticality for the ungauged
QED$_3$-GNY model. For the conjectured duality between chiral QED$_3$-GNY
critical point and deconfined quantum critical point, we find the inverse
correlation length exponent has a lower boundary $\nu^{-1}>0.75$, beyond which
the Ising-QED$_3$-GNY--$\mathbb{C}$P$^1$ duality may hold.
