See below tshuvos from HaGaon HaRav Asher Weiss shlit”a and HaRav Shlomo Aviner shlit”a on the subject. Both pasken not to say a Shehecheyanu on the vaccine.
Rav Asher Weiss - Vayigash 5781Rav Aviner Beracha on Vaccination
Shehecheyanu on Corona Vaccine?
Rav Aviner Beracha on Vaccination