New version of the question:
Given an odd dimensional manifold $V$, an almost contact structure is a pair of $(\alpha, \omega)$, where $\alpha$ is a non-vanishing 1-form and $\omega$ is a 2-form whose restriction to $\ker \alpha$ is non-degenerate. In particular, I don't ask that $\omega$ be closed. From this pair, I define a vector field $R$ with the properties that $\alpha(R) = 1$, and $\omega(R, \cdot) = 0$.
A class of examples come from hypersurfaces in a symplectic manifold. In those examples, the 2-form is the restriction of the ambient symplectic form (and is thus closed), and the 1-form is obtained by contracting the symplectic form with a normal vector field.
Suppose $V$ is a closed manifold. Can I deform the pair $(\alpha, \omega)$ so that the only minimal invariant sets of the resulting vector field are non-degenerate periodic orbits?
More generally, does this class of vector fields have any rigidity to it?
Explanation of the change: The first version of this question restricted attention to the class of examples of hypersurfaces in a symplectic manifold. From what Alvarez Paiva commented, it seems unlikely that this is doable. I am now allowing a larger family of deformations (since the 2-form is allowed to vary among maximally non-degenerate 2-forms, dropping the condition of being closed.)
Why I am asking this: I started thinking about this question in trying to understand some examples related to the question of putting a contact structure on an almost contact manifold. I don't think that my line of thinking is related to the original question anymore, but I am still curious to understand how soft almost contact is (compared to contact).