Suppose $M^n \hookrightarrow P^{n + k}$ is an isometric embedding of a manifold $M$ in an arbitrary ambient manifold $P^{n+k}$ for $k \geq 2$. Does there exist a global, non-vanishing section of the normal bundle, $NM$? I know that the Stiefel-whitney classes can provide obstructions to the existence of $q$ linearly independent sections of this bundle, but I'm unsure if there is a converse, even in the case of $q = 1$.
I would be interested in any relevant resources or known topological restrictions. If needed, one can assume that $M$ is simply connected, $P^{n+k} = \mathbb{R}^{n+k}$, and let $k$ be any positive number which allows for the construction of such a nowhere vanishing section.
To answer part of your question, there is no converse to the Stiefel-Whitney class' obstruction to the existence of a nowhere vanishing section. As a counterexample, we can take the zero section of the tangent bundle of the $2$-sphere $z:S^2\to TS^2$, a codimension 2 embedding whose normal bundle is isomorphic to $TS^2$. The Stiefel-Whitney classes of $TS^2$ are all trivial, but the normal bundle has no nonvanishing sections by the hairy ball theorem.