The ideal chain is the classic example of an entropic force. Usually one derives this force from the fundamental relation describing forces in the canonical ensemble: $$ \tag 1 F = (\partial \langle E \rangle / \partial x)_T - T (\partial S / \partial x)_T $$ where $x$ is the length of the chain. The first term is identified as ordinary force, and the second as entropic force. The definition of the ideal chain makes the energy $E$ constant for all configurations, and so there is no ordinary force. Yet, the entropy $S$ for a given $x$ varies (see the wikipedia article for why this is so), so there is an entropic force and we are told this is found to agree reasonably well with experiments.
There is an important connection with ideal gases. If we apply (1) to an ideal gas, with volume in place of $x$, the pressure of an ideal gas is seen to also be an entropic force, since its $\langle E \rangle$ is proportional to $T$ but independent of volume. So there is a strong analogy between stretching a rubber band (made of polymers sort of like ideal chains), and compressing an gas (made of molecules sort of like ideal gas particles).
However I have a doubt about the ideal chain model, that there is no apparent microscopic force acting in the ideal chain. Afer all, the same thermodynamic force can be written equivalently as: $$ \tag 2 F = (\partial \langle E \rangle / \partial x)_S $$ In an ideal gas, this expression of the force corresponds to to the microscopic picture of adiabatically changing the volume under the influence of molecules randomly pounding on its surface. Each time the boundary is moved inwards, there is a chance that a molecule is in the process of colliding with the wall and so the moving wall gives energy to the molecule. In this way when entropy is held constant, the energy of an ideal gas does change with volume.
The problem is that in an ideal chain, it seems impossible to understand the force using (2). We are told the energy is supposed to be constant which lets is ignore the first term in (1) but this would produce zero force in (2)! Analogizing to the ideal gas, there must be a microscopic force in a chain that is randomly pulling against the end of the chain, such that (2) agrees with (1), but what is it?
I have a strong suspicion about the solution, which I will add as an Answer below.