Is swearing not to know the suitor via the Internet, with an intermediary present, considered a false oath that necessitates a grave expiation?

Since you swore that you did not get to know the man online, and by that, you intended that you did not speak to him online, then what matters is your intention, not your family's intention. Therefore, there is no breach of oath upon you and no sin. You are not required to offer expiation for this oath.