Transfer files from mass storage device to other machine via wifi
You could buy an embedded Linux board with an SD card, a usb port, and a wifi card. Write a script that uses the rsync command to synchronize the contents of the SD card with your remote server over an SSH connection. Set the cron daemon to execute the rsync script once per minute (or maybe less frequently, depending on the speed of the connection).

You would have to generate an SSH key pair and put the public key on the remote server so that the board could login without a person having to type in a password.

For hardware, I'd try something like one of the following:


Hawkboard
Beagleboard
TS-7552