Err, remember, Muppets are still puppets. They're real *within* the movies and universe and so on, but they're made of fleece, foam, felt, or whatever. As the Muppet Eyes page notes, some characters do use glass taxidermy eyes for greater realism (see Eliot Shag or Jake the Polar Bear). But we shouldn't talk about organs, tissue, and so on when discussing behind the scenes aspects of Muppet design. That's getting too carried away. It's sort of like asking if Dagwood Bumstead's ink blot eyes actually somehow allow him to see out from the newspaper page.
Muppet organs are neither real nor fake. Puppets are made of cloth, and only appear to be alive because they are operated by humans. Were you to open up a Muppet, you would not find any organs. If, for whatever reason, the makers of the Muppets decided to show a Muppet organ -- say, to illustrate the workings of the brain, for example -- it would be done with cloth materials or computer generated imagery. Real organs would not be used for this process.
See eye movement. For Wembley, there's a mechanism to control the movement. The material used to make the eye doesn't matter, it's the mechanism that does the controlling (and with characters like the Gorgs or Ghost of Christmas Present, that's operated remotely through a waldo device. For Big Bird, the performer controls it all while inside the suit.
In some cases, like Floyd Pepper and Mahna Mahna, it's just a matter of the puppeteer folding down the eyes from within (since they're basically just sockets or circles which, when squeezed, cause the other side to come down and thus look like their eyes are closed or blinking). On others, like Kermit, the eyes are merely cosmetic or even absent but suggested via glasses (Dr. Bunsen Honeydew). Materials include cloth, foam, ping pong ball eyes as noted, fibre-glass, whatever. It doesn't really matter.