It seems that what you are asking is essentially how one can approximate the area of a surface via triangulations. This is a classical topic stretching back more than a century. The famous example of Schwartz Lantern shows that arbitrary triangulations do not work. Young first described conditions which triangulations should satisfy in order to approximate area:
W.H. Young, On the Triangulation Method of Defining the Area of a Surface, 1921.
These approximations have been studied extensively since then. For a description of Young's condition and other references see the following paper posted recently on the arXiv:
Kobayashi and Tsuchiya, Approximating surface areas by interpolations on triangulations, 2017.
See also
L. V. Toralballa, A geometric theory of surface area, 1970
for more references. I think that these papers answer your question. Some of these references are concerned only with 2-D surfaces, but Young's original paper seems to consider arbitrary dimensions. Further, these methods require very little regularity. I am pretty sure that the answer to your question is yes for surfaces in $R^3$, and I think that things should work in all other dimensions as well.