The following paper proposes a new target localization system design using an architecture based on reconfigurable intelligent surfaces (RISs) and passive radars (PRs) for integrated sensing and communications systems. The preamble of the communication signal is exploited in order to perform target sensing tasks, which involve detection and localization. The RIS in this case can aid the PR in sensing targets that are otherwise not seen by the PR itself, due to the many obstacles encountered within the propagation channel. Therefore, this work proposes a localization algorithm tailored for the integrated sensing and communications RIS-aided architecture, which is capable of uniquely positioning targets within the scene. The algorithm is capable of detecting the number of targets along with estimating the position of targets via angles and times of arrival. Our simulation results demonstrate the performance of the localization method in terms of different localization and detection metrics and for increasing RIS sizes.