A short summary of the relation between MPS/PEPS and ground states local Hamiltonians:
First the direction from MPS/PEPS to Hamiltonians:
Every MPS/PEPS naturally appears as the exact ground state of a frustration free local Hamiltonian. ("parent Hamiltonian")
For generic MPS/PEPS, this ground state will be unique.
There is a number of cases beyond the generic one where one can make statements about the ground state degeneracy. In particular, for translational invariant MPS the ground state degeneracy is always constant.
For translational invariant MPS, there is always a spectral gap above the ground state manifold; for PEPS, this holds only in certain cases.
Conversely, from Hamiltonians to MPS/PEPS:
Given a gapped local 1D Hamiltonian, its ground state is well approximated by an MPS. (cf. [Hastings '07] for the scaling)
Given a local 2D Hamiltonian where the density of states does not grow too quickly, its ground state (as well as thermal states) is well described by a PEPS (cf. [Hastings '05],[Hastings '07] for the scaling)
This concerns only the analytical relations between MPS/PEPS and Hamiltonians. In practice, better bounds concerning approximability etc. will typically hold.
Concerning "if and only if" relations, I think all these results only hold rigorously in one direction (though typically they might be "if and only if", e.g., I would think that typical gapless Hamiltonians will not have MPS ground states).
If you are looking for something more concrete, please let us know.