You need to use standard networking API. You can found more information about network programming for example here.

Amiga/MOS specific implementation of this API is in bsdsocket.library, provided by the TCP/IP stack. You can found more information about this specific implementation in the MiamiDX SDK, or here.

To retrieve a webpage, you need to use HTTP protocol. You can find more information about HTTP/1.1 protocol here.
You can also learn a lot from opensource HTTP clients, for eg. WGet. I also wrote a small HTTP retriever, which can parse the URL, and fetch files from a HTTP server. It's in PowerD and not C, but i can send it to you if you want it, just PM me.