Absolutely, the internet connection from your PC does not, in any way, allow the Pi to access the internet on it's own. The Pi should be considered as an individual computing device - unlike - for example - an Arduino. The Pi 'IS' it's own mini computer. Hence it needs it's own internet connection. It is ok connecting to the PC for SSH etc, but then it is on 'it's own and running the program it needs it's own net connection, just as another 'PC' would. That is exactly why it has it's own licence as it can act as a cheap standalone device running a monitor which is similar to a stand alone PC doing the same albeit not as powerful, but generally capable.
Let us know when you get it sorted
Joe