This paper addresses the problem of scalability for a cell-free massive MIMO (CF-mMIMO) system that performs integrated sensing and communications (ISAC). Specifically, the case where a large number of access points (APs) are deployed to perform simultaneous communication with mobile users and monitoring of the surrounding environment in the same time-frequency slot is considered, and a target-centric approach on top of the user-centric architecture used for communication services is introduced. In the paper, other practical aspects such as the fronthaul load and scanning protocol are also considered. The proposed scalable ISAC-enabled CF-mMIMO network has lower levels of system complexity, permits managing the scenario in which multiple targets are to be tracked/sensed by the APs, and achieves performance levels superior or, in some cases, close to those of the non-scalable solutions.