Transverse waves involve, as we all know, the displacement of particles of medium perpendicular to the direction of energy propagation. Thus, each element of the medium (in a very exaggerated, magnified picture) exerts shearing stress on the adjacent element. However, since fluids have no shear modulus to speak of, how is it that light (which is a transverse wave) propagates through water? Is it perhaps the electromagnetic character of light that permits this?
Light as a wave is not the same as wave motion in a collection of particles. Light passing through a medium does not displace particles in the medium itself. That would be a sound wave.
Rather, light is a wave where rather than the particles in the medium moving, the field lines of the electric and magnetic field move and oscillate. So the problem of having versus not having a shear modulus is not relevant to light.
It is indeed the electromagnetic character of light that permits this. Light propagates even in a vacuum because of Maxwell's equations. It can interact with a medium causing its speed to change* but the propagation isn't caused by the medium.
*The speed of light changes in materials but the speed at which information travels is always $c$.