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As Chris Gerig indicates in his comment, the correct notion is that of a contact-type hypersurface. These always exist, even locally. You could just take the boundary of a Darboux ball, which always exists. This is a contact-type hypersurface. Or, if you can find a Lagrangian submanifold (which you always can, e.g. a small torus in a Darboux ball) then the boundary of a Weinstein neighbourhood is a contact-type hypersurface. Or, if you are in a projective variety and can find an ample normal crossing divisor Poincare dual to the symplectic form then the boundary of a neighbourhood of the divisor is a contact-type hypersurface (see Seidel's "Biased viewpoint on symplectic cohomology").