I will present a brief survey of opinions here; much (digital) ink has been spilled on this topic so I can't claim this answer to be fully comprehensive, but it should provide an outline as well as resources for further study.
In this article (Hebrew) Rav Yosef Tzvi Rimon suggests a number of options:
- Ha-Tov Veha-Meitiv - thanking G-d for the vaccine's discovery and manufacture, leading to saving lives and a return to normality.
- Shehecheyanu (his preferred option) - thanking G-d for one's personal joy at receiving the vaccine.
- Alternatively (or in addition to one of the blessings listed above) the prayer brought in Shulchan Aruch Orach Chaim 230:4 to be recited before receiving medical treatment.
As well, he believes that one ought to say prayers of thanksgiving (Psalm 100, Psalm 30 or Nishmat Kol Chai)
In contrast, R. Asher Weiss in this article (Hebrew) does not recommend reciting either of the blessings suggested above. The custom (he believes) is to minimize their recitation in response to hearing good news, and otherwise to only recite them when seeing something new which makes the individual happy. In addition, the fact that the vaccine causes some initial discomfort is a further reason not to say shehecheyanu.
If one is in a state of such joy that he feels he must recite a blessing, R. Weiss recommends purchasing a new item of clothing and reciting shehecheyanu, with the intention of also applying the blessing to his joy at receiving the vaccine.
Finally, R. Hershel Schachter can be seen in this video recommending that one recites Ha-Tov veha-Meitiv (as he himself did); he has explained that this is the blessing "one makes on good things happening for the general public."