We continue investigating the generalisations of geometrical statistical models introduced in [13], in the form of models of webs on the hexagonal lattice $\mathbb{H}$ having a $U_{\mathfrak{q}}({\mathfrak{sl}}_n)$ quantum group symmetry. We focus here on the $n=3$ case of cubic webs, based on the Kuperberg $A_2$ spider, and illustrate its properties by comparisons with the well-known dilute loop model (the $n=2$ case) throughout. A local vertex-model reformulation is exhibited, analogous to the correspondence between the loop model and a three-state vertex model. The $n=3$ representation uses seven states per link of $\mathbb{H}$, displays explicitly the geometrical content of the webs and their $U_{\mathfrak{q}}({\mathfrak{sl}}_3)$ symmetry, and permits us to study the model on a cylinder via a local transfer matrix. A numerical study of the effective central charge reveals that for $\mathfrak{q}=-e^{i\gamma }$, in the range $\gamma \in [0,\pi /3]$, the web model possesses a dense and a dilute critical point, just like its loop model counterpart. In the dense $\mathfrak{q}=-{\mathrm{e}}^{i\pi /4}$ case, the $n=3$ webs can be identified with spin interfaces of the critical three-state Potts model defined on the triangular lattice dual to $\mathbb{H}$. We also provide another mapping to a ${\mathbb{Z}}_3$ spin model on $\mathbb{H}$ itself, using a high-temperature expansion. We then discuss the sector structure of the transfer matrix, for generic $\mathfrak{q}$, and its relation to defect configurations in both the strip and the cylinder geometries. These defects define the finite-size precursors of electromagnetic operators. This discussion paves the road for a Coulomb gas description of the conformal properties of defect webs, which will form the object of a subsequent paper. Finally, we identify the fractal dimension of critical webs in the $\mathfrak{q}=-{\mathrm{e}}^{i\pi /3}$ case, which is the $n=3$ analogue of the polymer limit in the loop model.