Is it permissible to give the vowed offering to the poor or to make expiation for it, after having inadvertently made a vow to one of the Imams, knowing that it is not permissible to make a vow to anyone other than Allah, and that the Imam's location is far away?

This vow is invalid and constitutes polytheism (shirk) with Allah, because it is an act of worship directed to other than Allah. Vowing is an act of worship. Allah Almighty says: (And whatever you spend of expenditures or make of vows—indeed, Allah knows it). And the Prophet, peace and blessings be upon him, said: (Whoever vows to obey Allah, let him obey Him; and whoever vows to disobey Him, let him not disobey Him).

Vows are not permissible except for Allah alone, because vowing is an act of worship. Prayer, sacrifice, vows, fasting, and supplication are all for Allah alone. He, the Exalted, says: (It is You we worship and You we ask for help). And He says: (And your Lord has decreed that you worship none but Him). And He says: (So invoke Allah, [being] sincere to Him in religion, even if the disbelievers dislike it). And He says: (And do not invoke with Allah anyone else).

You must repent to Allah from this, and you are not obligated to fulfill this vow.