Yeah, perhaps it is simpler to formulate this without transporting everything back via charts. If $M$ is an $m$-dimensional submanifold of $\mathbb{R}^n$, then $M$ in $\mathbb{R}^n$ locally looks like $\mathbb{R}^m\times\{0\}^{n-m}$ in $\mathbb{R}^n$.
The tangent bundles then look like $(\mathbb{R}^m\times\{0\}^{n-m})\times(\mathbb{R}^m\times\{0\}^{n-m})$ in $\mathbb{R}^n\times\mathbb{R}^n$ (with the usual canonical identifications), so the corresponding normal bundle is $(\mathbb{R}^m\times\{0\}^{n-m})\times(\{0\}^m\times\mathbb{R}^{n-m})$ in $\mathbb{R}^n\times\mathbb{R}^n$, which is an $…