The key word in the question is the coordinate neighborhood, so a clear definition of it would help.
Definition 1. A coordinate neighborhood of a point $p \in M$ is an open subset $U \subset M$ endowed with a collection of functions $x^i \colon U \rightarrow \mathbb{R}$, where $i=1,\dots,n$, such that the map
$$
\textbf{x} \colon U \rightarrow \mathbb{R}^n \colon p \mapsto
\left(\begin{array}{c}
x^{1}\\
\vdots\\
x^{n}
\end{array}\right)
$$
is a diffeomorphism onto its image. Briefly it is denoted by $(U,x^i)$. The functions $x^i$ are called the coordinate functions. The corresponding coordinate vector fields $\partial_i := \frac{\partial}{\partial{x^i}}$ are defined as the partial derivatives w.r.t. to coordinate $x^i$, so that $\partial_i x^j = \delta_i^j$. No Riemannian structure is involved so far.
There is so called the standard frame $(E_i \in \Gamma(T \mathbb{R}^n))$ in $\mathbb{R}^n$ such that $E_i = (0,\dots,1,\dots,0)$ with $1$ in the $i$-th position. The Euclidean metric $g^E \in \Gamma(S^2 T \mathbb{R}^n)$ is defined by
$$
g^E(E_i,E_j)=\delta_{ij}
$$
The coordinate frame $(\partial_i)$ is the pullback of the standard frame $(E_i)$ by map $\textbf{x}$, that is
$$
\frac{\partial}{\partial{x^i}} = \textbf{x}^*E_i
$$
Now, let $U$ be an open subset of a Riemannian manifold $(M,g)$. A smooth map
$$
F \colon (U,g|_U) \rightarrow (\mathbb{R}^n, g^E)
$$
is an isometry onto its image if $F_*g=g^E$ or, equivalently, $g = F^*g^E$. Recall, that for a diffeomorphism $F$ the pull-back is the inverse of the pushforward: $F^* = (F_*)^{-1}$.
As one can see from the question, the OP uses the following
Definition 2. A Riemannian metric $g$ on a smooth manifold $M$ is called locally flat if for any point $p \in M$ there is an open neighborhood $U$ of $p$ such that $U, g|_U$ is isometric to an open subset of $(\mathbb{R}^n, g^E)$. For brevity, the term "flat metric" is often used instead.
Let me restate slightly the fact in the question as the following
Proposition. For an open subset $U$ of a Riemannian manifold $(M.g)$ the following conditions are equivalent.
(i) $U$ is a "coordinate neighborhood" (of any of its points) in which the coordinate frame is orthonormal;
(ii) $(U,g|_U)$ is isometric to an open subset of $(R^n, g^E)$.
Proof.
$(i) \Rightarrow (ii)$ Check that map $\textbf{x} \colon U \rightarrow (R^n, g^E)$ provides the necessary isometry, i.e. $g = \textbf{x}^* g^E$. Indeed,
$$
g_{ij}=g(\partial_i,\partial_j)=\textbf{x}^* g^E(\partial_i,\partial_j) = g^E(\textbf{x}_* \partial_i, \textbf{x}_* \partial_j) = g^E (E_i, E_j) = \delta_{ij}
$$
which exactly means that the coordinate frame $(\partial_i)$ is orthonormal.
$(ii) \Rightarrow (i)$ Let $F: (U,g|_U) \rightarrow (\mathbb{R}^n,g^E)$ be an isometry. Define
$$
x^i (p) := F^i (p)
$$
i.e. $\mathbf{x} = F$. Now $(U,x^i)$ is a "coordinate neighborhood". QED.
As one can see, this is in fact a tautology: everything is hidden in the definitions!
Best Answer
Let $(M,g)$ be a Riemannian manifold. Fix $p \in M$ and choose a smooth local frame around $p$. By the Gram-Schmidt orthonormalization process, one can construct from the previous local frame a local orthonormal frame (by applying pointwisely the process). From the formula of the Gram-Schmidt process, this local orthonormal frame is smooth. Call it $(E_1,\ldots,E_n)$.
However, in order to be an orthonormaal coordinate frame, there must exist a coordinate patch $(x^1,\ldots,x^n)$ such that $E_j = \partial/\partial x^j$. As for a coordinate patch, the partial derivatives commute (that is, $[\partial/\partial x^i , \partial/\partial x^j]=0$, it follows that if $(E_1,\ldots,E_n)$ is an orthonormal coordinate frame, then $[E_i,E_j]=0$, which is a pretty rigid condition.
Moreover, if $(x^1,\ldots,x^n)$ is an orthonormal coordinate frame on $(M,g)$, it follows that the curvature tensor of $(M,g)$ vanishes on the neighbourhood where $(x^1,\ldots,x^n)$ is defined: this is because the metric in these coordinates is simply $(g_{i,j}) = (\delta_{i,j})$, that is constant, and hence, the Christoffel symbols are zero.
To conclude, I think you are misinterpreting what J. Lee is saying in his book. Given a fixed Riemannian manifold, there may not exist any orthonormal coordinate frame around a point. But of course, given any manifold with a coordinate system, one can construct a local metric for which the coordinate system is orthonormal.