As far as I know, libogc has no direct support for HTTP downloads. You'd have to write your own code to implement the HTTP protocol, or take existing code from other Wii Homebrew.

libogc only provides standard C functions like socket, sendto, ... that you can use to implement network communication.

No, they aren't. Most homebrew applications are open-source and almost all of them have some kind of HTTP code to check for updates or something.

USB-Loader GX (to download cheat codes and check for updates, github.com/wiidev/usbloadergx )
WiiFlow Lite (to download cheat codes and check for updates, github.com/Fledge68/WiiFlow_Lite )
Priiloader (to check for updates, github.com/DacoTaco/priiloader )
Homebrew Channel (to check for updates, github.com/fail0verflow/hbc )

and so on. Almost every Wii homebrew is open-source. Just make sure to follow the license if you take code from existing applications.