I am writing for a non-specialist audience, and I wish to make the claim that "every space endowed with a metric tensor has an orthonormal coordinate system." In other words, if, in $n$-dimensional space, we have a metric tensor $g_{ij}$, then we can find a coordinate system $(x_1,x_2,...,x_n)$ such that $$g_{ij} = \vec{e}_i\cdot\vec{e}_j = \delta_{ij},$$ where $\vec{e}_i$ is the vector associated with $x_i$.
Is it okay to make this claim, or are there caveats?
This is true at a point (such coordinates are called Riemann normal coordinates or geodesic normal coordinates [the former is a set of notes with details of the construction), but not generally true in an extended region, unless the curvature is zero (in a sense this is what curvature is: an obstruction to orthonormal coordinates staying orthonormal.)
One can also specify an orthonormal frame that remains orthonormal and varies smoothly along a smooth curve (as with the Frenet frame in 3D flat space), but again, curvature prevents one extending this to higher-dimensional objects because it prevents there being a single notion of parallel transport to from one point to another: if your space is curved and you parallel transport a vector along two different curves between the same two points, you in general get two different results.