The power over Ethernet (PoE)-enabled visible light positioning (VLP) networks as a promising technology can significantly enhance accuracy and costeffectiveness of indoor positioning. However, both the limited bandwidth of the light-emitting diode (LED) and the low sampling rate of the receiver have a negative impact on the positioning performance. Moreover, time synchronization requirements between transmitters and between transceivers become more stringent in a resourceconstrained VLP network. To address these issues, a PoEenabled VLP scheme with low bandwidth requirement and high-precision pulse reconstruction is proposed in the paper. Specifically, the precision time protocol and synchronous Ethernet are introduced to realize the synchronization transmission. Meanwhile, an on-off keying (OOK) modulation-based beacon signal is designed to unlock both the transceivers' synchronization and bandwidth requirements. Then, a high-precision pulse reconstruction method considering the LED model and impulse response is established to enhance the signal quality. Moreover, the position is estimated based on the maximum a posteriori (MAP) probability criterion. Experimental results obtained by the VLP testbed demonstrate that the proposed scheme outperforms the benchmark positioning schemes. It achieves a positioning accuracy of 1.7 cm by using the reconstructed 2 GHz sampling rate in the case of a bandwidth of 50 MHz and a real sampling rate of 100 MHz. Last but not least, the proposed scheme maintains positioning accuracy within 30 cm even with a few MHz bandwidth of LED.