Is it permissible to propose to a girl who writes comments on social media that amount to flirtation with her relatives, and is her action considered unlawful (haram)?

It is not permissible for a woman to flirt with a non-mahram (definition) man, whether verbally or in writing. What this girl is doing with her relatives is undoubtedly a reprehensible act, because it is a cause of fitna (temptation/strife). The Messenger of Allah, peace and blessings be upon him, said: "I have not left behind me any fitna more harmful to men than women."

As for marrying her: If the girl has repented, turned to Allah, and rectified her situation, or if she was ignorant of the ruling, then you may propose to her, because "the one who repents from a sin is like one who committed no sin." However, if she insists on this act, then look for someone else, for she is not pleasing in her religiosity. The Prophet, peace and blessings be upon him, guided us to marry a woman of deen (religious commitment), saying: "A woman is married for four reasons: for her wealth, for her lineage, for her beauty, and for her deen. So, marry the one with deen and you will succeed."