Yes, there is a link, both are examples of extremum principles. And yes it is possible to derive the principle of least action from the law of maximum entropy. The derivation is lengthy and I will only sketch the main steps needed:
We start from $dS \geq 0$ for an isolated system. For a non-dissipative system, this reduces to $dS = 0$. That is, entropy is a constant.
From the phase space structure we can show that the phase space state $\rho$ satisfies the equation $d\rho/dt = \partial\rho/\partial t - \mathcal{L}\rho$, where $\mathcal{L}$ is the Liouvillian.
From the constancy of entropy (1), we can derive the Liouville theorem $d\rho/dt=0$, which implies that the general equation of motion (2) reduces to the Liouville equation $\partial\rho/\partial t = \mathcal{L}\rho$.
For a mechanical system in a pure state, the phase space state is given by the well-known product of Dirac deltas; substituting this $\rho_\mathrm{pure}$ on the Liouville equation, the equation reduces to the Hamilton equations of motion.
Using the Hamilton Jacobi method, the Hamilton equations of motion can be written again as a single equation: the Hamilton Jacobi equation.
It can be shown that the Hamilton Jacobi equation "is an equivalent expression of an integral minimization problem such as Hamilton's principle", and Hamilton's principle is just the Hamiltonian version of the principle of least action. In other words, solving the Hamilton Jacobi equation one obtains the action $A$ [*] and this automatically satisfies the principle of least action $\delta A=0$.
Other versions of the principle of least action can be obtained from here. For instance, the Lagrangian version of the principle can be obtained using a Legendre transformation.
[*] References on mechanics usually denote the action by $S$, but here it would be confused with entropy.