The Pi does not do poe natively. POE is available as a hat, but I have not seen one for the Pi4 yet. The hats are nearly as expensive as the Pi itself.
I'm using the existing PoE hat on several Pi4's in a production environment and it works great. Never had shutdowns or issues with power. Note that I'm not drawing extra power from any of the USB ports, so perhaps the existing PoE hat isn't sufficient when you have USB devices attached to the pi4.
Good to know. I couldn't see any reason that they wouldn't work for a Pi4, but I couldn't find a single manufacturer that said Pi4 was actually supported.
Hat on the official site [1] explicitly says the 4B is supported.