  Recently, there has been significant interest regarding the regularization of
a $D\rightarrow 4$ limit of Einstein-Gauss-Bonnet (EGB) gravity. This
regularization involves re-scaling the Gauss-Bonnet (GB) coupling constant as
$\alpha/(D-4)$, which bypasses Lovelock's theorem and avoids Ostrogradsky
instability. A noteworthy observation is that the maximally or spherically
symmetric solutions for all the regularized gravities coincide in the $4D$
scenario. Considering this, we investigate the wormhole solutions in the
galactic halos based on three different choices of dark matter (DM) profiles,
such as Universal Rotation Curve, Navarro-Frenk-White, and Scalar Field Dark
Matter with the framework of $4D$ EGB gravity. Also, the Karmarkar condition
was used to find the exact solutions for the shape functions under different
non-constant redshift functions. We discussed the energy conditions for each DM
profile and noticed the influence of GB coefficient $\alpha$ in violating
energy conditions, especially null energy conditions. Further, some physical
features of wormholes, viz. complexity factor, active gravitational mass, total
gravitational energy, and embedding diagrams, have been explored.
