We study the universal critical properties of the QED$_3$-Gross-Neveu-Yukawa model with $N$ flavors of four-component Dirac fermions coupled to a real scalar order parameter at four-loop order in the $epsilon$ expansion below four dimensions. For $N=1$, the model is conjectured to be infrared dual to the $SU(2)$-symmetric noncompact $mathbb{C}$P$^1$ model, which describes the deconfined quantum critical point of the Neel-valence-bond-solid transition of spin-1/2 quantum antiferromagnets on the two-dimensional square lattice. For $N=2$, the model describes a quantum phase transition between an algebraic spin liquid and a chiral spin liquid in the spin-1/2 kagome antiferromagnet. For general $N$ we determine the order parameter anomalous dimension, the correlation length exponent, the stability critical exponent, as well as the scaling dimensions of $SU(N)$ singlet and adjoint fermion mass bilinears at the critical point. We use Pade approximants to obtain estimates of critical properties in 2+1 dimensions.
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