Fundamentally, it would be easy to do. 4 Port USB hubs normally require 2 Amps of 5v power. You can get 12v to 5v 2+ Amp supplies quite easily. They can be wired in parallel to a cigarette lighter. Cigarette Lighters are typically on their own 10 to 15 Amp fused circuits, and most are switched, so they are only powered when the car's ignition is set to ACC/On.
BUT from your question and comments, you have a 12v DC to 120v AC inverter, and plug in a home/office usb hub power supply. You COULD replace the cigarette lighter with the inverter, wiring it in. Some cars even come with that as a standard option. Just remember, this is atleast 120W of energy on the car side, and 120v at the inverter side, so wiring could be dangerous. Here is an example of a large 400W inverter wired in (cable to battery) and modded to the center console of a car. Smaller single port inverters could just replace the cigarette lighter without needing to run a cable to the battery, using the cigarette lighter fuse and it's own fuse as the protection.
That said, the three easiest, probably cheaper ways of doing it, would be getting a 2 amp 12v to 5v power supply and wiring that to your usb hub, or getting a 4 port usb car charger, or getting two 2 port usb car chargers for like 5 bucks and wiring those in. People Hardwire usb ports in their dash or center console all the time.
BUT the MOST IMPORTANT THING IS, making sure that the circuit you use, is switched. If it is not, that means the circuit is always powered, even with the car off and key removed. If you can use the cigarette lighter like that (car off, key out), you run the risk of the power supply or inverter you are using to drain your battery. Replacement car batteries arn't cheap, and at best, having to get jumped in the morning would ruin your day.
Normally AM radios use a magnetic loop (ferrite rod antenna) for picking up the H field part of the electromagnetic wave they are trying to receive: -
A closed box of steel isn't going to substantially prevent the H field penetrating inside. Think about the fact that you can extend the range of a magnet by putting some iron on one of the poles. If it were mu metal (rather than steel) the eddy currents set up might start to substantially reduce the H field of course.
An FM radio probably uses a short dipole (to be able to place it inside the chocolate box tin) and significantly this receives the electric field part of the EM wave so, the box is acting like a faraday cage and blocking the signal. A short dipole is also not going to naturally pick up anything like the sort of signal needed - typically a quarter wave at 100MHz is 0.75 metres and if choc boxes were that sort of size I'd be impressed!
Having said all of that, my answer is only justifying the facts as stated in the question. If someone asked what would happen if a radio were placed inside a choc box tin I'd still mention the H field and E field reception stuff but, intuitively, I wouldn't be able to make a call on it one way or the other.
Best Answer
You should always start by using devices that comply to decent EMC guidelines (which often rules out cheap garbage products you buy for 1 or 2 $). Then the next step is shielding the source of interference, but that only works when you actually have radiated interference. Conducted interference needs different approach eg. using power line filters or proper decoupling designed into the circuit.
Normally the antenna cabling in a car is already shielded / coaxial, so an extra shield there will not be very likely to work.
It is likely that your interference is conducted rather than radiated. This means that the device you plug in is doing bad things to the power supply lines, and the copper wire distribute this noise everywhere to all other devices in the car.