For two such functions $f$ and $g$, the difference $f-g$ is invariant under chip-firing, i.e. it factors through a function from the cokernel of the Laplacian to $\mathbb Z/k$. Conversely, for such a function $f$, adding any linear function from the cokernel of the Laplacian to $\mathbb Z/k$ produces another such function. 

So if there are any such functions, they are classified by linear functions from the cokernel of the Laplacian to $\mathbb Z/k$.

Such a function exists if and only if there do not exist integer vectors $v, w  \in \mathbb Z^n$ where $k v =\Lambda w$, for $\Lambda$ the Laplacian, and the sum of the entries of $w$ is nonzero mod $k$.

The "only if" direction is straightforward. Given such $v,w$, we would have $0 = kf(v)= f(kv) = f(\Lambda w) $ equal to the sum of entries of $w$.

For the "if" direction, write the cokernel of the Laplacian as a product of cyclic groups $\mathbb Z/n_i$ generated by vectors $v_i$. 
 Then $n_iv_i$ lies in kernel of the Laplacian, so we can write $n_i v_i = \Lambda w_i$ for some vector $w_i$. We can choose $f(v_i)$ such that $f(n_i v_i) $ is the sum of entries of $w_i$. We can do this as long as the some of entries of $w_i$ is divisible by $\gcd(n_i, k)$, which it is because $\Lambda w_i k/\gcd(n_i,k) = k ( v_i n_i / \gcd(n_i,k))$ and $v_i n_i / \gcd(n_i,k)$ is integral so the sum of entries of $w_i k/\gcd(n_i,k)$ is divisible by $k$ by construction. 

Having made this chose, we can define $f$ for an arbitrary $v$ by writing it as a integer linear combination of $v_i$ plus a vector of the form $\Lambda w$, and taking the appropriate linear combination of $f(v_i)$ and the sum of the entries of $w$. This is well-defined because the only ambiguity consists of adding $n_i$ to the coefficients of $v_i$, which we checked is consistent, and adding something in the kernel of the Laplacian, whose consistency follows from the assumption.