Your question is a little confusing, so let me give you a couple pieces of information.
First things first - the VIn pin on the Arduino is strictly for power going to the Arduino board, you shouldn't be drawing power from this pin. The documentation in the link you sent is a little misleading on this - it's basically saying 9v is coming OUT of the PoE module, and going IN to the VIn on the Arduino itself. If you wanted to tap into this 9v power, you should connect to the PoE module VOut instead.
PoE by default can deliver 37-57 volts or 42-57 volts, depending on what type of PoE you're using (which 802.3 specification). Wikipedia has a good overview of this information.
That being said, the Arduino Ethernet board you're using can't handle this voltage by itself - unless these boards have changed, they require an additional PoE module, which is basically just a DC-to-DC buck converter that will convert the high PoE voltage down to 9v or 12v. At this point, the Arduino's built-in converter can consume that 9 or 12v and distribute it at 5v across the board.
Now, to your problem - you want to power a relay that requires 7-12v. You could theoretically connect the relay to the VOut of the PoE module, giving you 12v. However, if the relay draws too much power for too long, you will either get brownouts (and your Arduino will restart) or potentially damage the Arduino board. Unless the relay draws a lot of current or is being powered for extended periods, you should be fine. If you want to be more thorough, you could put a large capacitor between your PoE module and the relay.
You should NOT try to connect the relay directly to the PoE VOut, as the voltage will be far too high.