Much attention has focused on algorithmic audits and impact assessments to hold developers and users of algorithmic systems accountable. But existing algorithmic accountability policy approaches have neglected the lessons from non-algorithmic domains: notably, the importance of interventions that allow for the effective participation of third parties. Our paper synthesizes lessons from other fields on how to craft effective systems of external oversight for algorithmic deployments. First, we discuss the challenges of third party oversight in the current AI landscape. Second, we survey audit systems across domains – e.g., financial, environmental, and health regulation – and show that the institutional design of such audits are far from monolithic. Finally, we survey the evidence base around these design components and spell out the implications for algorithmic auditing. We conclude that the turn toward audits alone is unlikely to achieve actual algorithmic accountability, and sustained focus on institutional design will be required for meaningful third party involvement.