Let $(M,g)$ be a smooth connected oriented $d$-dimensional Riemannian manifold. Let $S \subseteq M$ be a $k$-dimensional embedded submanifold which is compact, connected and orientable.

Suppose we are given a $W^{2,2}$ isometric immersion $F:(S,g|_S) \to (\mathbb{R}^d$,e) where $e$ is the Euclidean metric. ($F \in W^{2,2}(S,\mathbb{R}^d)$ and $dF_p \in O(T_pS,\mathbb{R}^d)$ a.e).

Let $NS \subseteq TM|_S$ be the normal bundle to $S$ in $M$.

Does there exist a section $q\in W^{1,2}(S;NS^*\otimes\mathbb{R}^d)$ such that $$ dF \oplus q \in \text{SO}(g,e) \, \,? $$ i.e we require $(dF_p \oplus q_p):(T_pM,g) \to (\mathbb{R}^d,e)$ to be an orientation-preserving isometry almost eveywhere on $S$. (We are looking for a "weak" section of $NS^*\otimes\mathbb{R}^d$).

If it helps, I am ready to assume $S$ is contractile or even a disk. (This question seems more analytic thant topological, see comment 2 below).

Partial results and comments:

1. In codimension $d-k=1$, the answer is positive, since pointwise, there is a unique choice of isometric completion $q$ (and this choice has the suitable regularity, as you can see here and here). In higher codimension some choices have to be made.

2. It is important that the question is asked in a weak regularity setting. Even if we knew $F$ was smooth, it is not true that we could always choose a smooth isometric completion $q$. Since bundles are isomorphic iff they are isometric, such a choice would exist if and only if the bundles $NS,\big(dF(TS)\big)^{\perp}$ were isomorphic. This would immediately imply that $TM|_S$ is a trivial vector bundle, which of course does not hold in general. However, even if $TM|_S$ is trivial, it does not imply the normal quotients are isomorphic. Of course, over a contractible space, all these topological delicacies disappear.

So, in some sense, we ask here if $NS,\big(dF(TS)\big)^{\perp}$ are "weakly isomorphic". (Though even if this is true, care needs to be taken regarding the regularity of the obtained isometry). Even though any two smooth bundles are weakly isomorphic, here $\big(dF(TS)\big)^{\perp}$ is not even a topological bundle, since its fibers don't change continuously.

A word on motivation:

This mini-problem arises in the context of determining the energy scale of thin elastic sheets ("thin manifolds"), in the language of Gamma-convergence. (I can elaborate more but this seems to much a digression).

Let $g$ be a smooth Riemannian metric on $\mathbb{R}^d$. Let $D=D^k \subseteq \mathbb{R}^d$ be the $k$-dimensional closed unit disk. ($k<d$).

Suppose we are given a $W^{2,2}$ isometric immersion $F:(D,g|_D) \to (\mathbb{R}^d$,e) where $e$ is the Euclidean metric; $F \in W^{2,2}(D,\mathbb{R}^d)$ and $dF_p \in O(T_pD,\mathbb{R}^d)$ a.e.

Let $ND \subseteq T\mathbb{R}^d|_D$ be the normal bundle of $D$ w.r.t to the metric $g$.

Question: Does there exist $q\in W^{1,2}(D;ND^*\otimes\mathbb{R}^d)$ such that $$ dF \oplus q \in \text{SO}(g,e) \, \,? $$ i.e we require $dF_p \oplus q_p:(T_p\mathbb{R}^d,g_p) \to (\mathbb{R}^d,e)$ to be an orientation-preserving isometry almost eveywhere on $D$.

In codimension $d-k=1$ the answer is positive, since $q$ is determined uniquely by $F$ (in a way that preserve the regularity).

In higher codimension some choices have to be made.

Motivation:

This mini-problem arises in the context of determining the energy scale of thin elastic sheets ("thin manifolds inside thick manifolds"), via Gamma-convergence. (Elaborating more seems a digression).