  The six-vertex model and its spin-$S$ descendants obtained from the fusion
procedure are well-known lattice discretizations of the SU$(2)_k$ WZW models,
with $k=2S$. It is shown that, in these models, it is possible to exhibit a
local observable on the lattice that behaves as the chiral current $J^a(z)$ in
the continuum limit. The observable is built out of generators of the su$(2)$
Lie algebra acting on a small (finite) number of lattice sites. The
construction works also for the multi-critical quantum spin chains related to
the vertex models, and is verified numerically for $S=1/2$ and $S=1$ using
Bethe Ansatz and form factors techniques.
