The VG Resource

I have no idea how I'm going to be able to rip Sprites from the SNES Mode of BizHawk.

Because for some reason, I can't seem to get to the Graphics Debugger because the feature is gray and disabled.

You're not emulating hard enough. :^)

You're using the snes9x core, which doesn't have graphics debugging support. Change the core to bsnes (it's an option within one of the drop-down menus of the emulator)