Erk. I don't have a blueprint or a diagram at the moment, but I can describe what I built for Hoppy. I used regular pine 1X2s, "ripped" into 1X1 lengths. The cart consisted of a square base that held the axles, and an open box frame that supported the sling. The base was built to be the same width as the frame, but four inches longer; I did this to make it a little more stable, and to provide a support for food and water cups. The frame was built separately from the base, and attached to it with bolts.
The frame itself needed to be just a bit wider and longer than the body of the bird; I think it was two inches wider and four inches longer. The height required had to be just enough to let him get his good leg underneath him, so I hung him in a sling at different heights until I worked out what would be best. I had to take the wheels into account, too, and also how much sag the new sling would have.
The base was put together with Liquid Nails and reinforced with wood screws (Liquid Nails is not strong enough at all). I then measured and drilled holes on each side so that I could thread two lengths of piano wire through; these formed the axles. I put on model airplane wheels that I'd gotten from a crafts store. If you have a model airplane store near you, go over and talk to them about your axle design. When they get done laughing over the idea of a cart for a chicken, they usually have some really good information and how-to pointers on building the axles and putting the wheels on.
The frame was also put together with Liquid Nails and wood screws. I made the sides an inch taller than the front so that Hoppy could lean down and get into his food and water. I also left the cross-pieces long on the top wall of the front and back; this gave me tie-down points for the harness and sling.
The sling was just two pieces of fleece fabric with four layers of roll cotton batting sandwiched inside, with leg holes cut out and hemmed. I installed strips of linen along all four sides so that I could adjust the hang of the sling and draw it tight or loose. Be sure to leave a cutout V in the back for poops to fall through; that was a problem I had with the first design.
The harness I put on Hoppy was a simple loop around the neck, tied down to the front of the cart, and two long pieces of linen that passed over his body on either side of the tail, went under the back bar, around the corners, and tied in back. They kept him from pushing himself up and out of the sling: when he pushed up, he carried the cart forward. He really had no use of one leg at all and pretty poor use of the other, so he wasn't coordinated at all.
Looking this description over, it's a little confusing. Please let me know if you'd like more detail.
Good luck with your bird!