There are manifolds without non-constant global meromorphic functions, such as generic K3 or a torus. Among these K3 surfaces, there are ones with (-2)-curves, which give a counterexample to the question. It is not hard to see that a K3 admits a -2-curve if and only if it has a vector with square -2 in its Picard group, and no non-constant meromorphic functions if its Picard lattice is negative definite. Global Torelli theorem implies that any sublattice of $H_2^3+(-E_8)^2$ is realised as a Picard lattice, allowing one to find such an example.
Edited the answer. If you require your subvariety to be not a divisor, there are still many examples of such, say, a product of two tori or to K3 K3 with no subvarieties and no global meromorphic functions. For a more interesting example, you may take a Lagrangian projective subspace in a hyperkahler manifold and deform the complex structures in such a way that this subvariety remains complex analytic, and no global meromorphic functions remain. This is explained in this paper: http://arxiv.org/abs/1401.0479
Edited again. There are subvarieties in complex manifolds admitting no
global holomorphic functions in any neighbourhood, though I don't know any examples for Kahler situation. Examples of such subvarieties (in fact, rational
curves) for twistor spaces are given in this paper: http://arxiv.org/abs/1211.5765