This is a good question! I couldn't find any reference for this, but here are my thoughts.
What does it mean to have a gapped / gapless boundary? It means that, keeping the bulk intact, we can tweak the Hamiltonian terms near the boundary and see what kind of spectrum it produces, comparing it to the case with no boundaries.
With this definition, it is trivial to see that the toric code (or any state/phase for that matter) can have arbitrarily complicated boundaries. We can always dress the boundary with a CFT, and this will give us "toric code with gapless boundary conditions". (1)
This is why it is useful to use the language "admits", as you highlight in your question. The interesting question to ask is "What is the simplest$^\text{a}$ boundary that this phase admits?" Of course we can make the boundary arbitrarily complicated regardless of the bulk phases, but some bulk phases do not allow for "simple" boundaries, and this is what is exotic about Symmetry protected topological phases (SPT) and Topological order (TO). In more technical language, general SPTs and TO induce an anomaly on its boundary, which implies that the boundary must necessarily be nontrivial$^\text{b}$.
The solution (1) for "does the toric code have a gapless boundary condition?" may be unappealing, so here is a more natural solution. Consider the toric code with smooth boundaries (I am using the convention $B_p = \prod X$). Add a boundary term $\Delta H_{lb} = -\lambda X_{lb}$, where ${lb}$ is summed over the qubits on the smooth edge. Note that $A_{vb}$, where $vb$ are vertices on the smooth boundary has nontrivial commutation with this boundary Hamiltonian. But all the other terms in the toric code Hamiltonian commute among themselves and with these terms so we can set them all to $+1$. Now let us focus on $\Delta H_{lb}$ and $A_{vb}$.
You can see that they have the same algebra as the terms in a transverse field Ising model$^\text{c}$. Therefore, there is a phase transition at $\lambda = 1$. Here, the boundary transitions from smooth to rough boundary (seen by setting $\lambda \to \infty$). The critical point corresponds to 2D Ising CFT on the boundary, and setting $\lambda = 1$ gives you toric code with gapless boundary conditions.
$^\text{a}$Simple usually refers to low central charge for gapless boundaries, low GSD for gapped boundaries. Gapped being simpler than gapless.
$^\text{b}$Naively, this is not true for the toric code since it admits a gapped boundary. But the boundary of the toric code is indeed anomalous in a certain sense, described in Schuster et al.
$^\text{c}$There is a caveat here. Strictly speaking, the effective boundary Hilbert space is the TFIM restricted to $\mathbb{Z}_2$ global symmetry singlet sector, but this does not affect the transition. Indeed, this restriction is related to the boundary anomaly and there is a good review in section II of Schuster et al.
