Moduli spaces of polygons have been studied since the nineties for their topological
and symplectic properties. Under generic assumptions, these are symplectic manifolds with natural
global action-angle coordinates. This paper is concerned with the quantization of these manifolds and
of their action coordinates. Applying the geometric quantization procedure, one is lead to consider
invariant subspaces of a tensor product of irreducible representations of $SU(2)$. These quantum
spaces admit natural sets of commuting observables. We prove that these operators form a semi-
classical integrable system, in the sense that they are Toeplitz operators with principal symbol the
square of the action coordinates. As a consequence, the quantum spaces admit bases whose vectors
concentrate on the Lagrangian submanifolds of constant action. The coefficients of the change of
basis matrices can be estimated in terms of geometric quantities. We recover this way the already
known asymptotics of the classical $6j$-symbols.