Let $(Y,\lambda)$ be a contact manifold, with a codimension-2 contact submanifold $(S,\lambda|_S)$ (this requires $TS\pitchfork\text{Ker}\lambda$). On $Y$ there is a natural vector field, the Reeb field $R$, determined by $d\lambda(R,\cdot)=0$ and $\lambda(R)=1$.

**Does there exist a contact form $\lambda'$ on $Y$ such that its Reeb field $R'$ is parallel to $S$, i.e. $R'|_S\subset TS$?**

This is true for $\dim Y=3$, as (in the compact scenario) $S$ will be a transverse knot. Using the Moser trick, there is a tubular neighborhood $N(S)\approx S^1\times\mathbb{D}^2$ with $\text{Ker}\lambda|_{N(S)}=\text{Ker}(dz+r^2d\theta)$. Thus $\lambda|_{N(S)}=e^H(dz+r^2d\theta)$ for some function $H:S^1\times\mathbb{D}^2\to\mathbb{R}$. The Reeb vector field on $(S^1\times\mathbb{D}^2,dz+r^2d\theta)$ is $\partial_z$ with Reeb orbit $S=S^1\times\lbrace 0\rbrace$. So $\lambda'=e^{-H}\lambda$ will do the trick.

In general, I am not sure what obstructions would arise by taking the Reeb field on $S$ and trying to extend it to the ambient $Y$.