At the American History Museum they have an Oscar the Grouch and a Sesame Street street sign on permanent display. They've had them for years - those are owned by the museum.
They have the Kermit (along with several other Muppets) in the "Jim Henson's Fantastic World" - these are on loan. I know that a Kermit was on display for a while, but I believe that's the same one that's currently with the traveling exhibit.
Now they own the original collection of Sam and Friends puppets. These were previously displayed on a "on loan" basis in 2006 (before the museum closed for an extremely lengthy refurbishment).
Muppet Labs is doing just fine... though I guess I should change my signature, I haven't worked there since 2007. Still with the company, just not at Muppetvision anymore. I'm not sure what you mean by a model Muppet Labs, though.