Majorana fermions are their own antiparticles, and Weyl fermions are just Majorana fermions without mass. However, I haven't been able to find any source that says whether a Weyl fermion is its own antiparticle.
My suspicion is that the question is meaningless. My impression is that "X is its own antiparticle" means "the mass eigenstates are mapped to themselves under charge conjugation". In the absence of a mass, we can choose any basis we want, so the question doesn't have a well-defined answer.
Then again, we can unambiguously pick out particles and antiparticles for a massless complex scalar field using the $U(1)$ charge. But I'm not sure if such a charge exists for Weyl spinors.
Is a Weyl fermion its own antiparticle? Generally, what does 'being your own antiparticle' mean?