Suppose that $M$ is a Riemannian manifold, and $p \in M$, $(x^i)$ is a normal coordinate of $p$. Then does $\{\frac{\partial}{\partial x_i}\}$ constitute an orthonormal frame of $M$.
I can show that at point $p$, $\{\frac{\partial}{\partial x_i}|_p\}$ is an orthnormal basis. But what about it being evaluated at other points in the normal neighborhood?
No, not unless $M$ is flat on the domain of the chart $(x^i)$ - the coordinate frame being orthonormal means exactly that $g_{ij} = \delta_{ij}$. In general the best you can do is "orthonormal to first order", since we have $$g_{ij} = \delta_{ij} + C R_{ikjl} x^k x^l + o(|x|^2).$$