Do Weyl fermions carry electric charge? Do Weyl fermions carry ordinary electric charge? That is, do they interact with, for instance, electrons or photons?
 A: We should probably start by pointing out that no Weyl fermion has ever been observed. The recent observations are of quasiparticles that behave like Weyl fermions. Speaking rather loosely (and at the risk of upsetting the QFT experts hereabouts) a Dirac fermion can be viewed as a sum of two Weyl fermions, and the observations are of paired quasiparticles obtained by splitting up electrons into the two Weyl components.
Anyhow, now we've got that out of the way the answer is that yes a Weyl fermion can carry electric charge and therefore can interact with photons and other charged particles. Weyl fermions are massless, but apart from this distinction a charged Weyl fermion would interact like any charged particle.
A: 
Do Weyl fermions carry electric charge?

That depends on whether Weyl fermions exist, and whether they are what people say they are. See this from the article mentioned by John Rennie:
'Whereas electrons and all the other known fermions have mass, in 1929, mathematician and physicist Hermann Weyl theorized that massless fermions that carry electric charge could exist, so-called Weyl fermions. “Weyl fermions are basic building blocks; you can combine two Weyl fermions to make an electron,” says condensed matter physicist Zahid Hasan at Princeton University.'
Note though that we make electrons (and positrons) out of photons in pair production. The photon has no mass or charge, the electron has both. Because charge and mass go hand in hand. You can't have one without the other. There are no charged particles that have zero mass. And there are no massive particles that don't have some kind of charge. Note that a neutron has its magnetic moment which betrays the existence of charge. There's no net charge, but there is charge. People might challenge this and point to say the neutral pion. But it has a mean  lifetime of 8.4×10$^{−17}$s and it decays via the electromagnetic force so it isn't a good example of mass without charge. Nor are quasiparticles. Quasiparticles aren't particles in the particle-physics sense.  

That is, do they interact with, for instance, electrons?

Photons interact with electrons, and photons don't carry electromagnetic charge. 

What about photons?

Photons interact with photons "via higher order processes". Check out two-photon physics and the Breit-Wheeler Process. 
But to get back to the issue: note this in Einstein's E=mc² paper: "If the theory corresponds to the facts, radiation conveys inertia between the emitting and absorbing bodies". You could contrive annihilation and pair production to destroy charged particles at one location A and create them at another location B. A massless photon doesn't carry charge, but mass and charge can be indirectly conveyed from A to B. And what other particles do we know about? What particle has properties that are closest to the photon? Google on Weyl fermions neutrinos and you can read things like "it was long thought that neutrinos were Weyl fermions". If I were you, I wouldn't rule them out. Neutrinos interact weakly with electrons. 
Note that a Dirac spinor is a bispinor, and that The Möbius strip also features in the Mathspages Dirac's belt article where you can read that it's "reminiscent of spin-1/2 particles in quantum mechanics, since such particles must be rotated through two complete rotations in order to be restored to their original state". There's two orthogonal rotations in a Möbius strip. For all you know there may be a way to "split" an electron to create two neutrinos flying apart in orthogonal directions. You might need a positron to do this or some other contrivance, but like I said, you can't rule it out. And then of course you could perform the reverse process at location B. 
