Do particles in a sound wave ever move transversally?

If you have any kind of solid material, it will become a little bit thicker as you compress it, and thinner as your stretch it. This means that a "one dimensional" wave traveling longitudinally down a rod will in fact cause some lateral motion. The ratio of displacements in the perpendicular direction is obtained from the strain (relative displacement of adjacent particles - how much local deformation there is) and the Poisson ratio of the material (a value between -1 and +0.5) that describes how much longitudinal strain turns into lateral strain.

So yes, it does move "the slightest bit".

That video is very poor in one aspect: particles in the sound field doesn't move "horizontally" nor "vertically". Really, the proper word is "longitudinal motion" and you are in fact asking about "transversal motion".

In basic description, the air is considered to be an ideal fluid. Therefore no shear stress is possible and hence no transversal motion as well.

In fact, there is a (little) viscosity of air, so yes, it's generally possible for particles to move a bit transversal as well. It could be observed especially in boundary layers on solid bodies. Of course, inside or on top of solid bodies or highly-viscous fluids, the transversal motion combined with longitudinal is natural.