  We study webs in quantum type $C$, focusing on the rank three case. We define
a linear pivotal category $\mathbf{Web}(\mathfrak{sp}_6)$ diagrammatically by
generators and relations, and conjecture that it is equivalent to the category
$\mathbf{FundRep}(U_q(\mathfrak{sp}_6))$ of quantum $\mathfrak{sp}_6$
representations generated by the fundamental representations, for generic
values of the parameter $q$. We prove a number of results in support of this
conjecture, most notably that there is a full, essentially surjective functor
$\mathbf{Web}(\mathfrak{sp}_6) \rightarrow
\mathbf{FundRep}(U_q(\mathfrak{sp}_6))$, that all $\mathrm{Hom}$-spaces in
$\mathbf{Web}(\mathfrak{sp}_6)$ are finite-dimensional, and that the
endomorphism algebra of the monoidal unit in $\mathbf{Web}(\mathfrak{sp}_6)$ is
$1$-dimensional. The latter corresponds to the statement that all closed webs
can be evaluated to scalars using local relations; as such, we obtain a new
approach to the quantum $\mathfrak{sp}_6$ link invariants, akin to the Kauffman
bracket description of the Jones polynomial.
