If light (a photon) is linearly polarized, say vertically, does it have some vertical spatial extent (perhaps in amplitude)?
The photon itself does not have any physical extent: it just travels along a line, as a point particle so far as we can tell.
It carries with it two fields: the electric field $\vec E$ and the magnetic field $\vec B$. These are both vector fields, in that they have vector values at every point in space. Off the trajectory of the photon, their value is the zero vector. At every point along the trajectory of the photon, however, you can associate a vector value for both the electric and the magnetic field. As a concrete example, take a linearly polarized photon traveling along the z-axis. The electric field oscillates in the direction of the x-axis, and the magnetic field oscillates in the direction of the y-axis. So light is often depicted in textbooks as having two oscillations around the axis of the trajectory, representing the electric and magnetic fields. The fields themselves are only present along the trajectory of the photon, though.
EDIT: The above description is for a single photon, or for an infinitely narrow beam of light. In practice, that doesn't exist. In practice (for example, a laser), you have a beam of light with some finite extent. The actual intensity of photons might vary with position across the beam of light. So in most experimental setups, there will be some spatial extent to your beam of light.