I think the best thing to start with is to give almost all the female parts to Miss Piggy. Maybe Donna Hayward can be played by Janice, the musician and nurse with the big lipstick. But a vast array of Miss Piggies in all pretty girl or crazy lady roles seems best, partly because Miss Piggy will kick our asses if we don't give her the pretty girl roles and will shine in the crazy lady ones. I mean, she IS Nadine, and she'll demand to be Laura Palmer. Josie will be a stretch for her, but I can't for the moment think of any Muppets who could easily do a very intense Chinese lady with close-cropped hair. Other casting by my own prejudices is below:
Dr. Jacoby: Dr. Teeth
Pete Martell: Oscar the Grouch
Sheriff Harry S. Truman: Fozzy Bear
The One-Armed Man: Animal
Benjamin Horne: Sam the Eagle
Backwards Talking Dwarf: Elmo
Deputy Andy Brennan: Grover
I'm having trouble casting Albert Rosenfield, Ed Hurley, Leo, Bobby Briggs, and a few others.
On the subject of Bob, the first thing that comes to mind is Animal, but Animal is far too innocent and bouncy despite the hair. I was thinking maybe Zoot or Sgt. Floyd Pepper from Dr. Teeth and the Electric Mayhem, but part of me just wants to go for the gold and give Gonzo the part.
Corrections and suggestions are more than welcome. For reference here's a list of Muppets and a Twin Peaks cast reference.