We study submanifold stabilization problems from an input-output perspective, where plant and controller are relations on their sets of input-output signals. In contrast to the classical input-output approaches, we consider signals whose integral p-fold distance to a submanifold is finite. For feedback interconnections of relations on such signals, we develop a framework to show that the distance of the output of the plant to the desired submanifold remains bounded. Within this framework, we present a small-gain theorem, a feedback theorem for conic relations, and a feedback theorem for passive relations. We connect our findings to multiplier theory and present applications to synchronization and pattern generation.