In Section 9.4 of S. Weinberg's book "The quantum theory of fields" it is shown how to get the Feynman $i\epsilon$-prescription in the propagator of a free scalar field using path integrals and explicit construction of the vacuum vector in the space of functionals on field (for the vacuum see formulas (9.2.9) and (9.2.12)).
It is claimed that there is a version of these constructions for fermions which leads to analogous Feynman $i\epsilon$-prescription in the propagator. The details are not presented. I am wondering about them. In particular I will be happy to see an analogous explicit description of the vacuum vector.
Since in the case of fermions one should use Berezin integration, there might be some subtleties.