Without cutting wiring apart? For clarification, I'm talking about the lighter-style port on the left and not the USB port on the right. I've googled and find later model instructions but it doesn't seem to pertain to the 2012.

It's a 2012 Z71 with the cheaper dash option:

 

No it's always on

 

Yes it is. The fuse for that power port is in the engine compartment. Determine which side is the hot sideand which is the load side.
Find a function that is switched and use that fuse socket to install one of those piggyback, 2 fuse circuit tap devices. Replace the original fuse for that circuit in the lower fuse slot. Add a small sized male spade connector (same size as the fuse leg) to the pigtail on the circuit tap device and plug it into the load side of the original power port fuse socket. Place the power port fuse unto the remaining slot in the circuit tap device.
Done.

 

Funny. My wife's big complaint on her Subaru is how the sockets don't work to charge her phone when the key is off, but in both our pickup trucks all those sockets always have power, even without a key in. I guess if you are concerned with draining the battery, making them switched would make sense.

I think @BrianGee has the best advice above, as it keeps any rewiring under the hood, and doesn't disturb anything under the dash.

 

No actual rewiring required. Just add this "Mini Add-a-circuit Fuse Holder". Available at any auto parts store for like $3.00.

 

I just bought a 12V socket that had a switch built in. I only use it for USB anyway so I kept the original socket to the right but replaced the old socket with a new switchable one. The bezel around the radio comes out without any screws so. I think I had to unscrew the heater/fan control panel but really didn't take more than 30 minutes. Here's a pic.


Switchable 12V USB outlet