What is the ruling on a sister bothering her sister who requested marriage before her, and cutting off communication with her?

It is not permissible for a Muslim woman to obstruct her sister's marriage, as this involves envy, which is forbidden in Islam. It is not a condition to delay the marriage of the younger sister until the older one marries; indeed, the marriage of the younger sister might even be a cause for the older one to marry. It is not permissible to abandon one's sister; rather, forgiveness and pardon should be sought, aspiring to Allah’s pleasure, in accordance with His saying, "But whoever pardons and makes reconciliation - his reward is due from Allah. Indeed, He does not like wrongdoers" (Quran 42:40), and His saying, "And not equal are the good deed and the bad. Repel [evil] by that which is better; then suddenly that between whom and you was enmity [will become] as though he was a devoted friend" (Quran 41:34). Also, according to the Prophet, peace and blessings be upon him, "Do not hate each other, do not envy each other, do not turn away from each other, and be, O servants of Allah, brothers. It is not lawful for a Muslim to abandon his brother for more than three nights."