No, the vaccine and antibodies the body produces after vaccination do not harm the formation of the placenta in any way. False claims that the vaccine contains a spike protein integral to the formation of the placenta and that the vaccine causes an immune response against said proteins have both been debunked.
A recent Northwestern Medical study of placentas from patients who received the vaccine during pregnancy found no evidence of injury to the placenta.
The Centers for Disease Control and Prevention recommends that people who are pregnant, trying to get pregnant, might become pregnant in the future, or breastfeeding get vaccinated and stay up to date with their COVID-19 vaccines. There is no evidence that the COVID-19 vaccine causes any problems with fertility or pregnancy, including the development of the placenta. You can read more of CDC’s COVID-19 Vaccines While Pregnant or Breastfeeding for the most up to date information