What is the ruling on a group of women reciting specific adhkar (remembrances) that include mentioning one of God's names a certain number of times, without raising their voices or engaging in collective supplication, in order to confront a calamity that has befallen one of them?

A Muslim is required to follow the Sunnah (definition) of the Messenger, peace and blessings be upon him, in his acts of worship. There is no mention of gathering together when a calamity befalls someone, nor of the manner described in the question, in any authentic hadith (definition). Therefore, you should abandon this practice and suffice with supplicating for her in her absence, and visiting her to comfort her and remind her of Allah, and that calamities are expiations for the believer.