You should specify what you mean by a polygon: a broken line or a surface. If we are talking about a broken line, then some classification is given here:
MR1703691 Kapovich, Michael; Millson, John J. On the moduli space of a spherical polygonal linkage. Canad. Math. Bull. 42 (1999), no. 3, 307–320.
If we are talking about surfaces, this is a different problem which is much more complicated. When the angles are sufficiently small, s that the whole polygon is a subset of the sphere, a classification is obtained in
F. Luo and G. Tian, Liouville equation and spherical convex polytopes, Proc. AMS, 116 (1992) 4, 1119-1129.
If the interior angles can be arbitrarily large, there is no known classification, even for the case of quadrilaterals. Some partial results can be found in
Eremenko, Gabrielov, Tarasov, arXiv:1405.1738 Metrics with conic singularities and spherical polygons
We actually have a classification of quadrilaterals up to isometry but it is not ready for publication yet.