The battery-less Internet of Things (IoT) devices are a key element in the sustainable green initiative for the next-generation wireless networks. These battery-free devices use the ambient energy, harvested from the environment. The energy harvesting environment is dynamic and causes intermittent task execution. The harvested energy is stored in small capacitors and it is challenging to assure the application task execution. The main goal is to provide a mechanism to aggregate the sensor data and provide a sustainable application support in the distributed battery-less IoT network. We model the distributed IoT network system consisting of many battery-free IoT sensor hardware modules and heterogeneous IoT applications that are being supported in the device-edge-cloud continuum. The applications require sensor data from a distributed set of battery-less hardware modules and there is provision of joint control over the module actuators. We propose an application-aware task and energy manager (ATEM) for the IoT devices and a vector-synchronization based data aggregator (VSDA). The ATEM is supported by device-level federated energy harvesting and system-level energy-aware heterogeneous application management. In our proposed framework the data aggregator forecasts the available power from the ambient energy harvester using long-short-term-memory (LSTM) model and sets the device profile as well as the application task rates accordingly. Our proposed scheme meets the heterogeneous application requirements with negligible overhead; reduces the data loss and packet delay; increases the hardware component availability; and makes the components available sooner as compared to the state-of-the-art.