I am sure it done somewhere, but I do not know a ref. I did something like this in my "Metric minimizing surfaces", but do not want claim originality.

You may fix a finite set of points draw all the geodesics between them.
Together these geodesics form a finite graph; they cut finitenumber of discs from your surface.
Exchange each disc by plane polygon provided by Reshetnyak's majorization theorem and you get an approximation.

By Reshetnyak's theorem, the constructed polyhedral surface admits a short map to the original one which is isometric on the finite set you started with.
From this convergence  follows, assuming you choose right notion of convergence for the surfaces.