You can absolutely prepare a photon in a linear polarization state. From the Wikipedia article you cited the very first line reads:
Photon polarization is the quantum mechanical description of the classical polarized sinusoidal plane electromagnetic wave. An individual photon can be described as having right or left circular polarization, or a superposition of the two. Equivalently, a photon can be described as having horizontal or vertical linear polarization, or a superposition of the two.
A linear polarization is just a superposition with equal amplitudes of circularly polarized light (and vice versa). For instance if you have a light beam you can send them through a polarizing beam splitter which splits the the output into light polarized horizontally or vertically relative to the beamsplitter itself (represented in the figure below as a square with a line through it). If you send a single photon through the beamsplitter than the photon can be detected using single photon detectors at each output (such as an avalanche photodiode or superconducting nano-wire detector) depending on it's polarization.
Even if the photon is unpolarized before the beamsplitter, it will be polarized afterwards. So if put two beamplitters in a row such that the horizontal polarized output from the first goes into the second beamsplitter (and you ignore all photons that go out the vertical port of the first beamsplitter), then 100% of the time photons that make it to the second beamsplitter will go come out the horizontal output, indicating that you have prepared a stream of single photons in a linear polarization state.
Note, you can detect linear polarization oriented along any axes (other than vertical and horization) by either rotating the beamsplitter or the beam. The standard way is to use a half-wave plate to rotate the state of polarization relative to the orientation of the waveplate (which is in a mount you can rotate to whichever angle you want). Circular (and eliptical) polarizations can also be measured by adding a quarter-wave plate to this setup as well and oriented as desired/needed.
A schematic of this setup borrowed from this paper is shown in image below.