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!