打 is the verb used for any kind of injection: 打针 (get a shot), 打点滴 (get an IV drip), 打疫苗 (get vaccinated). The 打 here means 'to administer/strike-into', not 'hit'.
culture
Standard childhood vaccines are managed via a 预防接种本 (vaccination booklet). The COVID era made 打疫苗 part of daily small talk, with phrases like 第三针 (the third dose) and 加强针 (booster shot) becoming common.
