  Based on work done by Bonechi, Cattaneo, Felder and Zabzine on Poisson sigma
models, we formally show that Kontsevich's star product can be obtained from
the twisted convolution algebra of the geometric quantization of a Lie
2-groupoid, one which integrates the Poisson structure. We show that there is
an analogue of the Poisson sigma model which is valued in Lie 1-groupoids and
which can often be defined non-perturbatively; it can be obtained by symplectic
reduction using the quantization of the Lie 2-groupoid. We call these
groupoid-valued sigma models and we argue that, when they exist, they can be
used to compute correlation functions of gauge invariant observables. This
leads to a (possibly non-associative) product on the underlying space of
functions on the Poisson manifold, and in several examples we show that we
recover strict deformation quantizations, in the sense of Rieffel. Even in the
cases when our construction leads to a non-associative product we still obtain
a $C^*$-algebra and a "quantization" map. In particular, we construct
noncommutative $C^*$-algebras equipped with an $SU(2)$-action, together with an
equivariant "quantization" map from $C^{\infty}(S^2)\,.$ No polarizations are
used in the construction of these algebras.
