If you want plug and play here are the only ways you can do this in the way you explained.
This is bypassing any computer and letting the device do it itself.
1) Microcontroller, Atmel AVRs can do this with the right setup. You will either need to go the WiFi route or Ethernet. If Ethernet you will need to plug it into a hub or switch or into the router, but you will need the proper hardware. If ethernet, then an ethernet controller and proper setup, check out the e ehternet shield for arduino.
Likewise, if wifi, you will need hardware for that too, wifly shield or other shield for arduino to get the idea. Now, if wif, it will need to be hhardcoded into the micrrocontroller the login info for the network, or have a simple touch interface or other means of interface to tell the microcontroller the login stuff. But this is more than doable. I've done it multiple times, after the microcontroller has a connection, either ethernet or wifi, test it with a ping to google, if okayed, send a GET or POST request to said site, hell could even make your own protocol and listen on some random port for numbers or somthing, it is all up to you.
2) RaspberryPi or similar embbed OS type device (Note: This is extreme overkill for this). A GumStix would be much much smaller the a Pi, but what ever. Same principal, your either going to need an ethernet connection or wifi. If you want plug and play with nothing attached but power, then it will have to be wifi, and then you will need to have the creds to login to the wifi and shit if you can start to get what I'm saying here.
3) have something on your PC that you install. You wouldn't even need to install anything. Store a program on the device itself that you can run from the device you plugged in. All it would need to do is "barrow" the PCs internet connection to send the numbers or whatever. IDK what the application is, but if all you wanted to do was send a few numbers from a USB device to a specified server, then storing the numbers and the program to send it on the USB device would be simple and a lot cheaper than supplying a wifi radio on the device. Once the device is plugged in, it can so an OS check, if windows, run the EXE, if linux run the bash script or something. Or go the HID route and try do it VIA HID. the Teensy(google it) would be perfect.
Any questions, I'll get to them in time. Work is murdering me, so I'll answer whenever.