Simulations of photoneutron generation are presented for the anticipated experimental campaign at ELI-ALPS using the under-commissioning e-SYLOS beamline. Photoneutron generation is a three-step process starting with the creation of a relativistic electron beam which is converted to gamma radiation, which in turn generates neutrons via the $\left(\gamma ,n\right)$ interaction in high-Z material. Electrons are accelerated to relativistic energies using the laser wakefield acceleration (LWFA) mechanism. The LWFA process is simulated with a three-dimensional particle in cell code to generate an electron bunch of 100s pC charge from a 100 mJ, 9 fs laser interaction with a helium gas jet target. The resultant electron spectrum is transported through a lead sphere with the Monte Carlo N-Particle (MCNP) code to convert electrons to gammas and gammas to neutrons in a single simulation. A neutron yield of $3\times {10}^7$ per shot over $4\pi$ is achieved, with a corresponding neutron yield per kW of $6\times {10}^{11}$ n/s/kW. The paper concludes with a discussion on the attractiveness of LWFA-driven photoneutron generation on high impact, and societal applications.