Cardboard: not good (will get damp, moldy, collapse, host vermin) and unless you put in a WHALE of a lot of it, you won't get much R-value out of it anyhow.
Packing peanuts: have some potential if nothing else is available, but you can probably do better. Main problem is they don't have terrific R-value and will make an unholy mess if mice get in and/or there is an accident when you're doing the construction.
If you ask around at stores, you can often score a bunch of styrofoam sheets, ideally plain flat ones but molded-to-fit-merchandise ones can be used too. They get 'em as packing amterial and often are happy not to have to pay to have them hauled away as garbage. If you use styrofoam (peanuts or sheets) make REAL REAL SURE to have totally tight carpentry, b/c if mice can find a quarter-inch crack or gap to sneak in, they will set up a mouse empire inside the walls and track little tiny gravelly bits of styrofoam all over creation. What a mess!
Another possibility if you want very, very inexpensive materials is to pack TOTALLY MOISTUREPROOF walls with 4-6" inches of well-packed sawdust or shavings; or stack haybales or plasticwrapped shavings bales along the walls. All of the above can host mold and/or mice problems, but done right you can often get away with it to good effect, especially the version where you stack bales along the OUTSIDE of the coop walls. Shovelling snow up against the coop walls helps too.
Ultimately though the most effective insulation would be batts (you can sometimes get them secondhand) or rigid foamboard. The former is cheaper but more aggravating to work with. Insulate as heavily as you can -- if you can stagger out studs to make 6-8" insulated walls you will get a LOT less heat loss than a standard 4" stud wall.
Personally I do not think it is a good idea to use a vapor barrier in a chicken coop under most situations, btw.
Something you COULD use large sheets of cardboard for, btw, is if you decide you want to make a hover or roost-box or 'coop within a coop' partitioned off for the depths of winter to better hold the chickens' heat around them at night. Cardboard stapled to a light wood frame would work just fine for that.
Good luck, have fun,
Pat