Most general version for the Gauss-Bonnet theorem for polygons

There are no other conditions, and in fact a more general statement is true. The standard reference is the survey of Reshetnyak, Two-dimensional surfaces of bounded curvature, in the book:

MR1263963 Geometry. IV. Nonregular Riemannian geometry. A translation of Geometry, 4. Translation by E. Primrose. Encyclopaedia of Mathematical Sciences, 70. Springer-Verlag, Berlin, 1993.

The original source is the book of Aleksandrov and Zalgaller, Two-dimensional manifolds of bounded curvature, Trudy Mat. Inst. Steklov. 63 1962. There is an English translation.

Gauss Bonnet is stated and proved there in much larger generality: for "Aleksandrov surfaces of bounded curvature".