Evidence does not support this claim.
Trial data shows no decrease in rate of pregnancies or increase in reate of miscarriages.
The original clinical trials did not include pregnant people and participants were asked to avoid becoming pregnant, however there were numerous accidental pregnancies. Accidental pregnancies ocurred equally between vaccinated and control group, and miscarriages ocurred at a lower rate in vaccinated group when data is excluded from countries where a termination would be counted as a miscarrage for legal reasons (otherwise, the rate of miscarriage is similar between groups).
Explainer: https://drive.google.com/file/d/1_wHIYX-tGkGBPwuax7N8BxZPR4PTTCDm/view
Original (AZ): https://www.thelancet.com/journals/lancet/article/PIIS0140-6736(21)02282-0/fulltext
Original (Moderna): https://www.fda.gov/media/144434/download
Original (Pfizer): https://www.fda.gov/media/144246/download
Original (Janssen): https://www.fda.gov/media/146217/download
"Both COVID-19 and the vaccine elicited anti-COVID IgG antibodies that were detected in the FF [follicular fluid] at levels proportional to the IgG serum concentration. No differences between the three groups were detected in any of the surrogate parameters for ovarian follicle quality."
https://academic.oup.com/humrep/article/36/9/2506/6345462
"From our findings, the vaccine appears to have no impact on women’s fertility."
https://www.medrxiv.org/content/10.1101/2021.05.30.21258079v1
"No difference was found in serum hCG documented implantation rates or sustained implantation rates between the three groups. Reports claiming that COVID-19 vaccines or illness cause female sterility are unfounded."
https://www.sciencedirect.com/science/article/pii/S2666334121000684
For IVF treatment 7–85 days after vaccination, "No in-between cycles differences were observed in ovarian stimulation and embryological variables before and after receiving mRNA SARS-CoV-2 vaccination."
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8116639/
There has been concern that similarities between Syncytin-1 protein in the placenta and the spike protine in SARS-CoV-2 and vaccines might cause infertility, however this claim is not supported by evidence.
"Based on an EMBOSS water pairwise analysis, the overall amino acid identity and similarity between Syncytin-1 and the SARS-CoV-2 spike protein are only 8.8% and 15.8%, respectively. Moreover, SARS-CoV-2 spike is much larger than Syncytin-1... We did not detect crossreactivity between anti-SARS-CoV-2 spike protein antibodies and Syncytin-1."
https://www.nature.com/articles/s41423-021-00773-x
"BNT162B2-vaccinated women did not transmit vaccine mRNA to breast milk, and did not produce a concurrent humoral response to syncytin-1, suggesting that cross-reactivity to syncytin-1 on the developing trophoblast, or other adverse effects in the breast-fed infant from vaccine mRNA ingestion, are unlikely."
https://www.medrxiv.org/content/10.1101/2021.05.23.21257686v1
"No evidence of fetal defects or anti-syncytin-1 antibody induction following COVID-19 mRNA vaccination"
https://www.biorxiv.org/content/10.1101/2021.12.07.471539v1
UK Royal College of Obstetricians and Gynaecologists
"women trying to become pregnant do not need to avoid pregnancy after vaccination"
https://www.rcog.org.uk/en/guidelines-research-services/coronavirus-covid-19-pregnancy-and-womens-health/covid-19-vaccines-and-pregnancy/covid-19-vaccines-pregnancy-and-breastfeeding/
US CDC
"COVID-19 vaccination is recommended for all people 5 years and older, including people who are pregnant, breastfeeding, trying to get pregnant now, or might become pregnant in the future."
https://www.cdc.gov/coronavirus/2019-ncov/vaccines/safety/monitoring-pregnant-people.html