The 1D result is discussed in detail here: 1607.06766. (See also 1607.06504.)

The analog of the TVBW invariant is the FHK state sum, which takes as input a separable algebra $A$. As Kevin points out, this algebra has an interpretation as the space of states on the interval with some boundary conditions. The algebra has a canonical special symmetric Frobenius form, which is used to build the invariant. Not all 2d TQFTs, which are classified by commutative Frobenius algebras, have an FHK state sum; but up to deformation (appropriate for gapped phases), they all do. The construction is also redundant, as the TQFT only senses the Morita class of $A$. In particular, the space of states on the circle (the vector space underlying the commutative Frobenius algebra) is the center of $A$.

The analog of the LW model is the fixed-point Matrix Product State (MPS) system, which also takes as input a separable algebra. It consists of a parent Hamiltonian, whose ground states are have a simple wavefunction representation called MPS. For each module $V$ over $A$, there is a ground state $\vert\psi_V\rangle$ on the circle; the space of ground states is spanned by the indecomposable modules. Therefore, as was the case for the TQFT, the space of ground states is the center of the separable algebra $A$.

The correspondence is made more explicit by extending the state sum to incorporate physical (brane) boundaries. These are associated to modules $V$ over $A$. If one evaluates the state sum for the anulus with boundary condition $V$ on one boundary circle and outgoing cut boundary on the other, one obtains the MPS state $\vert\psi_V\rangle\in A^{\otimes N}$ before projecting to the TQFT state space (the center of $A$). The parent Hamiltonian arises as the projector assigned to the cylinder.

I agree with Kevin's opinion that the 1D result only becomes interesting when new structure is added. This is due to the trivial algebra $\mathbb{C}$ being the only indecomposable separable algebra in $\text{Vect}$. See the references above for systems with finite group actions. There, one works with separable algebras in $\text{Rep}(G)$ and has a Morita class of algebras for each pair $(H,\omega)$, an unbroken symmetry group and a cocycle characterizing the SPT order. For the correspondence between 1D fermionic systems and field theories that are sensitive to spin structure, see 1610.10075.