The short answer is that the vortex shedding is induced by an asymmetric perturbation of the flow resulting in an oscillating flow at frequency that is not constant but instead changes with the characteristic flow velocity. The structure has to be designed in such a way that the frequency of vortex shedding meets the eigenfrequency of the structure (it does not occur automatically). The following sections give a more rigorous explanation.
Vortex bladeless technology
Vortex bladeless - very similar to a wind turbine is able to extract energy from a moving fluid - which in this case is based on harvesting energy by aero-elastic resonance caused by the Strouhal flow instability and the development of a Karman vortex street that establishes for Reynolds numbers $Re := \frac{U L}{\nu} \gtrsim 50$ around cylindrical structures. In order to harvest the lift force that is generated by the oscillating flow, the structure has to be designed in such a way that the displacement is high and the oscillation continuous - so the structure's resonant frequency should be chosen close to the operating frequency. As a result the structure should be made of a material with a great fatigue resistance and low internal damping such as carbon fiber reinforced polymers.
This means the structure and material have to be tuned to the (average) wind velocities. A correlation between the characteristic length, characteristic velocity and force shedding frequency for oscillating flow is given by the dimensionless Strouhal number $St := \frac{f \, L}{U}$. So you could use the experimental curves for the Strouhal over Reynolds number for a rigid cylinder determined by Roshko (1954), Lienhard (1966) Achenbach (1981), the Reynolds number formed with the average and peak velocity of your location, the kinematic viscosity of air and the characteristic length corresponding to the diameter of your cylindrical structure, to determine an estimate for the vortex shedding frequency and design the stiffness of your system to have a eigenfrequency close to this operating frequency or design a tuning system that changes rigidity according to those curves for self-synchronisation using a simplified equation of motion (e.g. simple torsional oscillators, a short paper on this can be found here), numerical simulation and/or experiments.
Development of vortex shedding
I already gave an answer on the different flow regimes for the flow around a fixed cylinder in another post. The mechanism should be the same for the case of an oscillating cylinder as when exposed to a flow it will only start oscillating after vortex shedding occurs.
The cylinder moves the fluid to the sides inducing a certain vorticity. The basic idea is that inertia is dominating for increasing Reynolds number - the flow won't follow the curvature of the cylinder and will stall: A flow separation occurs and the wake behind the cylinder will be characterised by very low velocities and high pressure (Bernoulli's equation). The viscosity of the fluid will lead the diffusion of momentum into the wake and lead to small standing vortices behind the cylinder with a different direction of rotation.
This symmetrical flow is not be stable: Small perturbations in symmetry might lead to vortex shedding. I don't think the mechanisms for this are yet completely understood but here two possible explanations:
My former teacher of aerodynamics, who conducted experiments by means of slowly rising velocity (thus Reynolds number) explained it this way. The "pressure mountain" (the standing vortices) behind the cylinder hinders the particles to proceed, slows them down and even reverses their macroscopic motion. This leads to ternary vortices (pointing away from the cylinder) on either side of the cylinder and due to some asymmetric perturbation in velocity this would wash one of the vortices way inducing the vortex shedding.
I have only simulated flows around cylinders by means of numerical methods. From my experience with low Reynolds number numerical experiments the instability starts behind the cylinder due to a Kelvin-Helmholtz shear flow instability. Even for Reynolds numbers that should be characterised by vortex shedding (such as $Re = 100$) depending on the position of the boundary conditions and the numerical scheme, the flow would be symmetric with standing vortices for a longer amount of time. I would have to simulate for an extended period of time until the flow would trigger an instability from the back of the domain, the tail would start to oscillate. The two reconnecting streams seem to act like shearing flows and any small difference in velocity upstream (e.g. due to local numerical dissipation in my case) or any imperfection in geometry would lead to a small velocity difference between the two shearing layers. This results in a Kelvin-Helmholtz instability explanatory video here) which in combination with the vorticity induced by the cylinder will lead to the vortex street. Just look at this video of a numerical simulation of a fixed cylinder to see what I mean.
Potentially the mechanisms for this depend on the Reynolds number under consideration which are a lot higher in experiments with air (fully turbulent) than the ones that are typically considered by numerical experiments without turbulence models. In any case after the first vortex is shed, the initially vortex free flow has to stay circulation-free (circulation theorem) and thus a circulation with different orientation is introduced that leads to a separation on the other side of the cylinder. (You could also think of it as a flow that is blocked by the vortex, as soon as the vortex is washed away the flow will accelerate on that particular side and the pressure will be lower.) The resulting oscillating vortex-shedding flow, the von Karman vortex street, restores the symmetry (that you might expect from the geometric symmetry) in a statistical sense while a periodic force will be exerted. If the cylinder isn't fixed this will induce a motion.
I added two pictures I obtained by a 2D lattice-Boltzmann simulation from the same time step for $Re=100$. The upper one shows vorticity magnitude isolines (dashed: clockwise, solid: counter-clockwise), the lower one pressure (solid line: higher than inlet due to stagnation, dashed: lower than at the inlet).
