Spoiler for after a few minutes of thought

The sum is 0. It's pretty easy to see that for any right triangle the sum of such vectors is 0. Any triangle can be subdivided into 2 right triangles and the vectors from the subdivision line will cancel each other out, so every triangle has the sum of such vectors equal 0. Every convex polygon can be subdivided into triangles, so 0 for every convex polygon.

Now, a question... Does a polygon even need to be convex? I think this works for any polygon.