Is announcing that one of the sisters is in need of prayers due to illness or difficult circumstances, and the other sisters praying for her in a dedicated Islamic women's forum, considered an innovation (bid'ah) or a form of collective supplication (du'a jama'i)?

The prohibited collective remembrance (dhikr) is when attendees recite a litany or dhikr in a single voice, or when a leader recites something and the attendees repeat it after him. The request for prayer mentioned in the question is not part of this prohibited dhikr. The Prophet, peace and blessings be upon him, used to command seeking forgiveness for the deceased and asking for their steadfastness after burial. He also mourned the Negus and commanded seeking forgiveness for him. What the sisters do by requesting prayers is permissible, and it indicates goodness and helping those in need. The Prophet, peace and blessings be upon him, encouraged praying for a Muslim in their absence.