Let $M\subset N$ be a embedded submanifold and $\dim M\le \dim N-2$. Can we find a non-vanishing vector field on $M$, that is a vector field $X\in\Gamma(TN|_M)$ such that $X(p)\notin T_pM,\forall p\in M$
I think in this way we can find a larger submanifold $M\subset M'\subset N$ such that $\dim M'=\dim M+1$
Assuming the normal bundle of $M$ in $N$ is oriented (and rank $2$, for argument), you'll have a nonvanishing section precisely when the Euler class vanishes. (Perhaps the easiest example where that fails is $S^2\subset S^2\times S^2$, embedded as the diagonal.)
I also don't see how you plan to use a nonvanishing normal vector field to create $M'$ (unless you intend a manifold with boundary, in which case you can take a ribbon built by the vector field). But consider the example $\Bbb RP^2\subset\Bbb RP^3\subset\Bbb RP^4$. The normal bundle of $\Bbb RP^2\subset\Bbb RP^4$ is nontrivial, but we still get the scenario with $M\subset M'$ with $M'$ boundaryless in this case.