Is Sufism a bid’ah (innovation), since the word isn’t found in the Qur’an?

The absence of its name from the Qur'an does not make Sufism a bid'ah: many essential Islamic sciences, Tafsir, hadith science, theology, got their names only after the first generation. A thing is not bid'ah if the Shariah itself supports its purpose.

The objection is that Tasawwuf is a later invention because the term isn't in the Qur'an or Hadith. But by that logic, Tafsir (Qur'anic exegesis), Usul al-fiqh (jurisprudence), and the sciences of hadith would all be innovations too, their names also came after the first generation.

The scholars of Shariah held that bid'ah pertains to ends, not means. Imam Shafi'i said, "Anything which has a support (mustanad) from the Shariáh is not Bid'a, even if the early Muslims did not do it" [al-Ghimari, Tashnif al-adhan 133].

Tasawwuf exists to preserve something the Shariah commands, sincerity, and the states of the heart such as fear of Allah, love of the Prophet ﷺ, and freedom from envy and pride. Since the Shariah commands the purification of the self, the science aimed at it cannot be a reprehensible innovation.