Suppose $M$ is an embedded sub-manifold of $D$-dimensional Euclidean space $E^D$, with embedding $\phi:M \hookrightarrow E^D$.
And suppose I know the induced Riemmanian-metric $g$ on $M$, which depends by construction on $\phi$ (which so happens that I don't explicitly know).
Can I express $\phi$ in terms of $g$ (closed form or as a PDE maybe?).
No, it is not possible in general. A cylinder $S^1 \times \mathbb{R} \hookrightarrow \mathbb{R}^2 \times \mathbb{R}=\mathbb{R}^3$ is locally isometric to a flat plane $\mathbb{R}^2 \hookrightarrow \mathbb{R}^2 \times \mathbb{R}=\mathbb{R}^3$. So one can find a small piece of the cylinder and of the flat plane which are isometric to each other.
But their embeddings are quite different, for example the extrinsic curvatures on the cylinder can be nonzero, but on the flat plane are zero.