The authority, exactly
§ 71 AufenthG splits it: the missions abroad are competent for visa matters, and the Ausländerbehörde is competent for residence-law decisions inside Germany. The Opportunity Card is a residence permit under § 20a(1) AufenthG, so the Ausländerbehörde takes it over once you are there. The consent of the Federal Employment Agency is not required for the job-search card itself.