In this talk, we first review the notion of local symplectic groupoid integrating a Poisson manifold. We provide a general existence result for underlying generating functions and a general analytic formula when the manifold is a coordinate space. We then show that the formal Taylor expansion of this generating function for $t\pi$ at $t=0$ recovers an extract of Kontsevich's star product formula restricted to tree graphs, recovering a formal generating function introduced by Cattaneo-Dherin-Felder. Finally, motivated by the heuristic path integral expression for the star product, we introduce a system of PDEs and show that the non-formal generating function can be recovered by evaluating the PSM action functional (with 'sources') on its solutions. This provides a functional, non-perturvative construction of the Lie theoretic structure integrating the Poisson manifold in terms of the underlying PSM and we comment on underlying gauge-theoretic aspects.