Is it possible to create protons from photons? I've read a few pages on related topics (e.g., here) but not found this question.
Is this type of reaction theoretically possible that we know of?
I know pair production creates electron and positrons, so could there be an analogous mechanism for proton and antiproton?
I guess because protons are composed of three quarks the process would be much more complex.
 A: Yes , it is possible to create protons from photons , in gamma gamma colliders.

Photon beams can be made so energetic and so intense that when brought into collision with each other they can produce copious amounts of elementary particles

A pair can be produced from the energy of the two gamma photons.
It has already been seen here, with off mass shell gammas,
The feynman diagram is


The reaction $e^+e^−\to e^+e^−p \bar p$ is studied with the L3 detector at LEP.

A similar diagram is implied, and the data are  compared to models.
Of course in proposed collider only the part of the Feynman diagram with the gamma gamma,on shell, is relevant.
A: Yes. This is possible from colliding real photons (i.e., on-shell photons). I try to show you how this is possible within the context of Standard Model of elementary particles.
Protons ($p$) are composite particles composed of three quarks: two up quarks ($u$) of charge $+2/3$ $e$ and one down quark ($d$) of charge $-1/3$ $e$. While, anti-protons ($\bar p$) consist of two up antiquarks ($\bar u$) of charge $-2/3$ $e$ and one down antiquark ($\bar d$) of charge $+1/3$ $e$. So, in principle, production of $p \bar p$ is allowed in $γγ$ collisions. This can be visualized by the following Feynman diagram (hereafter, $q$ stands for quarks and $\bar q$ for antiquarks):

In the above diagram, the quark-quark-photon and antiquark-antiquark-photon vertices can be understood via electromagnetic interaction like QED. If the initial $γγ$ collision has enough energy for producing a proton-antiproton pair, then the process can lead to the production of a $p \bar p$ pair. For this purpose, we need three quarks (two $u$ quarks and one $d$ quark) and three antiquarks (two $\bar u$ antiquarks and one $\bar d$ antiquark). So we need such a Feynman's diagrammatic expansion:

The dashed part of this diagram consist of infinitely many terms according to the allowed vertices in electroweak and QCD theories (in the Standard Model of elementary particles). In QCD, there exist three vertices and I restrict myself here to QCD interactions. For example, one of the lowest Feynman diagrams can be illustrated as

Another example is this one:

Note that, I've implemented the four possible vertices (within QED and QCD) in this answer, summarized as

Now, we have ended up with three quarks ($q$) and three antiquarks ($\bar q$). Supposing that two of quarks are $u$ quarks and one of them is $d$ quark, and supposing that two of antiquarks are $\bar u$ antiquarks and one of them is $\bar d$ antiquark, finally, the three quarks hadronize into proton and the three antiquarks hadronize into anti-proton. This process has been confirmed experimentally and analyzed before in a number of studies. In conclusion, it is possible to create protons (with anti-protons) from photons.


You may also find the following (relevant) studies/papers useful:

*

*L3 Collaboration, "Proton–antiproton pair production in two-photon collisions at LEP", Phys. Lett. B 571, 11 (2003), arXiv:hep-ex/0306017.


*M. Kłusek-Gawenda, P. Lebiedowicz, O. Nachtmann, and A. Szczurek, "From the $\gamma\gamma\to p\bar p$ reaction to the production of $p\bar p$ pairs in ultraperipheral ultrarelativistic heavy-ion collisions at the LHC", Phys. Rev. D 96, 094029 (2017), arXiv:1708.09836.


*TASSO Collaboration, "Exclusive proton-antiproton production in two-photon collisions." Phys. Lett. B 108, 67 (1982).


*VENUS Collaboration, "Measurement of the proton-antiproton pair production from two-photon collisions at TRISTAN." Phys. Lett. B 407, 185 (1997).
