The Internet of Things (IoT) plays a fundamental role in monitoring applications; however, existing approaches relying on cloud and edge-based IoT data analysis encounter issues such as network delays and high costs, which can adversely impact time-sensitive applications. To address these challenges, this paper proposes an IoT framework called Sazgar IoT. Unlike existing solutions, Sazgar IoT leverages only IoT devices and IoT data analysis approximation techniques to meet the time-bounds of time-sensitive IoT applications. In this framework, the computing resources onboard the IoT devices are utilised to process the data analysis tasks of each time-sensitive IoT application. This eliminates the network delays associated with transferring large volumes of high-velocity IoT data to cloud or edge computers. To ensure that each task meets its application-specific time-bound and accuracy requirements, we employ approximation techniques for the data analysis tasks of time-sensitive IoT applications. These techniques take into account the available computing resources and optimise the processing accordingly. To evaluate the effectiveness of Sazgar IoT, experimental validation has been conducted. The results demonstrate that the framework successfully meets the time-bound and accuracy requirements of the COVID-19 citizen compliance monitoring application by effectively utilising the available IoT devices. The experimental validation further confirms that Sazgar IoT is an efficient and scalable solution for IoT data processing, addressing existing network delay issues for time-sensitive applications and significantly reducing the cost related to cloud and edge computing devices procurement, deployment, and maintenance.