First off, there are two USB ports on the Yun.
- A micro-USB port that is connected to the '32U4 processor (the one that runs the sketch.)
- A full size USB port that is connected to the Ar9331 processor (the one that runs Linux.)
Port #1 is a device port: you can connect it to a host device like a computer, but you cannot connect another device to it. Using that port, when connected to a computer, your Yun can appear like a serial port, a keyboard, or a mouse. There is no native support for it to act like a mass storage device. There is the LUFA library that supposedly can make the Yun appear to the computer as if it were a mass storage device, but from the little I've read about it, it's difficult to implement and slow. I would also imagine that it takes up a lot of the limited code space you have in a sketch.
Port #2 is a host port: you can connect devices to it like keyboards, thumb drives, web cameras, etc, but you cannot connect it to another host like a computer.
You say the USB port is available outside of your enclosure, and that it has an "antenna" connected to it. It can't be strictly an antenna, that makes no sense. It must be some sort of device like a WiFi interface (which would be redundant since there is already one on the Yun) or a cellular radio, or some other sort of radio. If so, I'm assuming you are talking about the large Type A USB port that is the host port on the Linux processor. I don't know of a way to connect that port to another computer so that the computer thinks it is a mass storage drive.
Perhaps a solution is to plug a USB thumb drive into that port? You could do some Linux programming to look for new devices connected to the USB port, and if it's a mass storage device it could make a copy of the SD card data onto the USB drive. If you have some sort of user interface on your device, and can initiate a copy that would give you more control, but you should be able to set it up to automatically copy whenever a USB drive is mounted. That way, the idea would be to unplug your "antenna" and connect a USB drive to back up your data. Then, you would remove the USB drive and connect it to your computer to transfer the files. Basically, a kind of sneaker-net.
Getting the Yun to recognize the USB drive should be easy - just plug it in and should be automatically mounted.
Copying the files should be easy - it's just a shell script full of copy commands.
Detecting when the USB drive has been inserted and removed, and calling the copy script will be the trick. I'm sure it can be done, I just don't have enough Linux experience, especially with handling USB devices, to help with that part.
Because the SD card is handled by the Linux processor, and the USB host port is also handled by the Linux processor, adding this ability shouldn't really affect your sketch, it is really just a bunch of Linux programming.