Arduino Data to Cloud without shield []
Without a shield? Yes, it's possible.

You send it serially to your computer as you are doing now. Then you write some software for your computer to read that data serially and send it on to your chosen web service.

Can the Arduino magically send data to the internet by itself without you investing in a network connection of some form (WiFi shield, Ethernet Shield, etc)? No, of course not. If it could there would be no market for the network shields, and so they wouldn't exist.