After looking at this question for a few days in the context of the Riemann curvature tensor, holonomy for a given affine connection, and the (false) conjecture that the parallel transport around the boundary curve could equal the integral of the Riemann tensor within the span of the closed curve, I've concluded that the Stokes theorem cannot be applied to this conjecture except when the connection is flat.

The reason for the failure of the conjectured relation between curvature and parallel transport is that the Stokes theorem's integrals are themselves not really well defined. But it is not that simple. I'll explain...

When even constructing a simple Riemann integral from the fundamentals, one has to add vectors at different points inside the region. Even if you have a connection, you have to decide which paths to use to connect the points of the region. You can do a kind of a "raster scan" of the image of a rectangular region of $\mathbb{R}^2$, parallel transporting the vectors back to the left of the scan to add them on the left hand side, and then you can parallel transport all of these X-scans down the Y-axis by transporting them down to the bottom left of the rectangle. But then what do you have? It's clearly not geometrically meaningful. And then you have the same problem with the boundary integral of a vector function.

A second conclusion which I came to is that if you do apply the Stokes theorem to this scenario, you get a mathematically correct identity, which has a practical value as the first iteration of the Picard iteration procedure to compute the parallel transport around the curve. This, clearly, is not extremely useful. But in my opinion, the Stokes theorem is applicable to this situation. It just doesn't give anything geometrically meaningful for a non-flat geometry, and it has limited value for relating curvature to parallel transport and holonomy. On the other hand, it does get the right answer in the limit of a shrinking region to a point, which gives the correct answer for the "Cartan characterization of curvature".

This issue is related to questions 16850 and 50051.