Is it permissible to hack an inactive Instagram account dedicated to posting Quranic verses with the intention of continuing the work on it, so that it becomes an ongoing charity (Sadaqah Jariyah) for its original owner and for the hacker?

Hacking accounts is a forbidden aggression in Islamic law, and its perpetrator is sinful. Therefore, it is not permissible for you to hack this account at all. What you mentioned about the account's nature, activity pattern, and its cessation does not permit you to hack it.

And Allah knows best.