  Recent sign-problem-free quantum Monte Carlo simulations of (2+1)-dimensional
lattice quantum electrodynamics (QED$_3$) with $N_f$ flavors of fermions on the
square lattice have found evidence of continuous quantum phase transitions
between a critical phase and a gapped valence-bond-solid (VBS) phase for flavor
numbers $N_f=4$, $6$, and $8$. We derive the critical theory for these
transitions, the chiral $O(2)$ QED$_3$-Gross-Neveu model, and show that the
latter is equivalent to the gauged Nambu--Jona-Lasinio model. Using known
large-$N_f$ results for the latter, we estimate the order parameter anomalous
dimension and the correlation length exponent for the transitions mentioned
above. We obtain large-$N_f$ results for the dimensions of fermion bilinear
operators, in both the gauged and ungauged chiral $O(2)$ Gross-Neveu models,
which respectively describe the long-distance power-law decay of two-particle
correlation functions at the VBS transition in lattice QED$_3$ and the
Kekul\'e-VBS transition for correlated fermions on the honeycomb lattice.
