What make/model laptop do you have? You need to be clearer on what you asking for.
The answer to your subject question is NO. If you are using a modem to connect the internet (via dial-up) you don't need a network controller.
Most laptops, however, use either a wireless or wired network controller to connect to the internet now. It is MUCH faster than a modem/dial-up.
All you should have to do is go to your laptop vendor's webite, like dell.com/support for example, and download and then install the needed device drivers for your make/model of laptop (including the webcam if built-in and any network adapters).
Please clarify and good luck!