Vaccination typically begins at 6 - 8 weeks (the age range when maternal antibody protection begins to break down), then boostered 1 - 2 more times at 2 - 3 week intervals. (web-dvm.net)
While babies do receive passive immunity from breast milk antibodies, the protection occurs inside the baby's gastrointestinal tract only — babies are not able to absorb these antibodies (rebelliousjezebel.com)
Like a flu or polio vaccine, a shot designed to protect against Zika would need to contain some form of the virus in order to stimulate antibody protection against it. (scientificamerican.com)