I believe that SS must be informed of you FIL's death with a copy of the death certificate from your MIL. Did they have a joint account? Does anyone have Durable POA for your MIL?
The funeral home generally takes care of these matters & files the necessary paperwork to the SS admin. Get numerous official (notorized) copies of the death certificate as they will be needed to switch names on accounts & file for death benefits etc.