打 (dǎ) is the verb used for any kind of injection: 打针 (dǎ zhēn, get a shot), 打点滴 (dǎ diǎndī, 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 预防接种本 (yùfáng jiēzhòng běn, 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.
