Is it permissible to chat online with a religious person (a relative or otherwise), given that the conversation is limited and pertains to religious matters, and taking into account the questioner's vow that she will not resume chatting with him?

Speech between a man and a non-mahram woman is permissible when there is a need, provided that privacy (khalwa) is avoided and there is no softness in speech (khudu' bil-qawl). As for speech without need, it is a door to temptation and may be a luring from Satan. If you vowed not to speak to a relative and refrained from speaking to him, then there is nothing upon you. However, if you resume speaking to him, it is more cautious to offer an expiation for an oath (kaffarat yamin).