Radio frequency (RF) energy harvesting is a promising technology that enables self-sustainable wireless Internet of Things (IoT) networks. In this article, we analyze the energy harvesting performance of a Wi-Fi-based IoT network, where a large number of IoT devices are connected via Wi-Fi for both data communication and energy transfer. Applying probability theory and statistical geometry, we first develop an analytical model to study the energy sustainability of wireless IoT devices with Wi-Fi charging, which operate in an active/charging mode and access the channel using carrier sensing multiple access with collision avoidance (CSMA/CA) protocol. Based on the analysis, we derive the necessary and sufficient conditions for the AP beaconing frequency and the charging period of IoT devices to ensure that a network with a general random topology is long-term energy sustainable. It is shown that transmission collisions due to random access result in too much energy consumption that makes it difficult to achieve energy sustainability of IoT devices. To maximize the total network throughput while ensuring long-term energy sustainability of Wi-Fi IoT devices, a distributed energy-sustainable throughput-optimal algorithm is proposed for user charging period selection. Extensive simulations using NS-3 validate the analysis and demonstrate the efficiency of the proposed algorithm.