This wasn't free to build, but depending on what you find, you may be able to make it for close to free. I spent about $100, total.
This instructible is more of a "duh, why didn't I think of that?" kind of thing, rather than something really creative. But my kids were impressed, so I decided to share.
Step 1: Supply List
Small battery operated, motion activated light
Rubber door mat with "nubs"
Step 2: Put things in and check fit
I decided to cut my door near the back of the box's right side. The litter box (I have one of those large ones) sits all the way to the left inside the box.
Step 3: Cut a cat door
the deck box I purchased has vertical stripes of two different textures, so I used two of them to assist in my vertical cuts. Start by just scoring your desired cut, then go back over your scored line with more pressure. Two or three passes should have you pretty well through the plastic of the box. The only difficulty will be where there is a thicker stability bar molded in - just take your time and you'll be through those thick parts in just a few more passes.
After making my two vertical cuts, I again used the box's decorative textures to decide where to cut. In the case of my box, the hole came out to be 5" wide by 8" tall. I wasn't sure that would be enough for my plump cat, but she walked in just fine, as did my tall cat.