Thanks everyone. You are all so good.
As for the questions... I'm using the word "permanent" loosely, not in the "has a foundation" sense like you see in most town codes. Because (as one poster mentioned) we're already illegal, there's not much use in me doing research on the codes... I'm already in the wrong! I need to go in and kiss butt. Basically, the town hall folks are suggesting ways to make the chicken situation more palatable to the zoning board (who will decide on the variance). Even a tool shed or play house where we have it would be considered illegal. Moving it, as we originally planned to do, doesn't seem enough for them. Actually, what the animal control lady told me was to find a "dog house", something small and unassuming that won't ruffle any feathers on the board. Yes, it all sounds nebulous and undefined. That's because it is. sigh.
Putting the whole coop on skids is certainly a good idea, but in the board's eyes, it will still look like a big ole shed. I'm not putting four hens in a dog house either, but another guy in zoning said a chicken tractor would definitely make the board more comfortable.
Will definitely check out the portable options suggested here. Thanks all for your kindness; it is very appreciated.
I am in Putnam County, NY, for the poster who asked. And to the other person who asked if the neighbors had any annoying pets... well, their many cats are when they scream at our door for food! But more annoying is the music they blare from their very un-zoned party room right on our property line. They aim the speakers at our house whenever they're annoyed at us, or have their teenager scream out the windows when we have company. We played nice for a long time... sending over veggies from the garden or eggs, etc, thinking we could win them over. We're not combative. But they don't care; they torture everyone around here. Now we've had enough, and called Town Hall about their party room. They're in big, big trouble, especially since it has illegal septic.
It's not rewarding to me though... only because I know they'll be looking for other ways to punish us. I wish we could move.