Let $N$ be a $(n+1)$-dimensional Riemannian manifold and $M\subset N$ a Riemannian hypersurface (embedded or immersed). Let $M$ and $N$ be oriented and choose a unit normal vector field $\nu$ along $M$ such that a basis $(b_1,\ldots,b_n)$ of $T_pM$ is oriented iff the basis $(\nu,b_1,\ldots,b_n)$ of $T_pN$ is oriented.
Now let $(b_1,\ldots,b_n)$ be a local orthormal frame for $M$, defined on some open set $V\subset M$ (V is open in $M$).
Does in this case always exist a local orthonormal frame $(\tilde{\nu},\tilde{b_1},\ldots,\tilde{b_n})$ for $N$, defined on some open subset $U\subset N$ (U is open in $N$) with $V\subset U$ and $\tilde{b_i}|_V=b_i$, $\tilde{\nu}|_V=\nu$. (After maybe shrinking $V$ to a smaller open subset of $M$.)
I am also interested in how to show that statement if it's true.
Just extend the frame arbitrarily to an open subset $U\subset N$, and then apply Gram-Schmidt to it. The Gram-Schmidt process won't change the vector fields on $U\cap N$ where they're already orthonormal.