Can neutrons be synthesized purely from protons and electrons?
Note: I'm looking for reactions that do not require neutrinos or any particles besides just protons and/or electrons as reactants.
Yes, they can be. The production of neutrons by deep inelastic scattering of electrons on protons was studied in HERA experiments in the 90's. No neutrino needed, at least not as one of the colliding particles. Here are some links:
"Free" electrons and protons can combine to form neutrons (and electron neutrinos) through weak interactions. The process would usually be energetically unfavoured in low density situations because the rest mass of the neutron is about 1.29MeV higher than the combined rest mass energies of the proton and electron.
This reaction, between free protons and electrons might occur in neutron stars. The reason is that Fermion degeneracy increases the energies of the electrons and protons, such that their sum can equal or exceed the Fermi energy of the neutrons.
But there is a snag. It turns out that if all the reactants are at their Fermi energies then it becomes impossible to simultaneously conserve momentum. See What allows the modified Urca process to work at lower density than direct Urca in neutron star cooling?
To get around this requires bystander particles (usually more neutrons) and so this would count as an additional reactant.
The direct process can occur at the very high temperatures inside the collapsing cores of supernovae, where the neutrons and protons are not yet fully degenerate, and is a vital cooling process (because the produced neutrinos may escape).