Is it permissible for the Imam of the mosque to ask the congregants to supplicate for a sick or injured brother or one in need during their prostrations (sujud), before or after the establishment of the prayer?

Supplication (Du'a) during prostration is permissible. Accordingly, it is permissible for the Imam or anyone else to ask the congregants to supplicate for a sick person's recovery or for another need. There is no harm in this, whether it is before or after the iqamah (call to commence prayer). However, after the iqamah, it is better to focus on the matter of the prayer.