When doing my fic-verse, I went to a LOT of trouble trying to iron out the wrinkles in "continuity", from Muppets being evolved from the talking creatures in Dinosaurs until now. I think I settled on the Muppets were cared for at some point by Nanny, but later forgot about it. Also, the cartoon, with its focus on imagination, seems to be something like an interpretation (after all, the dating would be ALL wrong, since Baby Rowlf liked punk rock and that didn't exist at the implied timeline ... the cartoon is heavily 80s but can't possibly be). Notice that in MFC, it's stripped down a bit more. You only see a single corner of a dimly lit room. There's nothing aside from the fact it's on film to date it. I tried looking at 8mm on Wiki, but it doesn't address (unless I missed it) the fact you can hear them singing, which is weird to me because all of my grandfather's vids were silent. This is assuming, of course, that the Muppetverse is close to reality technology-wise. I get the impression it's implying they were babies in the 50s, but that seems odd considering Kermit isn't a kid at that time, right? I'm thinking of Sam&Friends, as I've never seen Kermit's Swamp Years. So, as there might be no official "canon", I suspect there are enough common elements (such as Piggy's poor rural background or Kermit's, well, "swamp years") to suggest a possible canon backstory.
edit: And trying to fit Fraggle Rock and other properties is a major league headache, LOL. Fraggles don't keep accurate dating records (all we have to go on, it seems, are throw-away lines by Gorgs) and the fact that the Rock's tunnel systems are magically somewhat independent of time and space make pinning down its history dang near impossible.