Short answer: Yes, Cat5e can do PoE (Power over Ethernet).
There are three ways a cat5e cable will deliver PoE: Alternative A, Alternative B, and 4PPoE.
Alternative A transports power on the same wires as data for 10 and 100 Mbit/s ethernet variants.
Alternative B transports the power on separate pairs, meaning the data and power are separated into 2 different pairs, making troubleshooting easier.
4PPoE provides power on all four pairs of twisted cables. This enables the cable to provide power for higher power applications.