Unless things have changed since Nov. of 1999, you don't have to have any special documents for domestic animals....ie: dogs and cats other than current complete vaccination records from a veteranarian and stamped by the USDA.
I took both my cats to the vet....got their vaccines.....the vet filled out a form for both cats.....and then I sent it to the nearest USDA for a stamp including a $16 US fee. The vet can provide you a contact address and telephone# for the USDA office nearest you.
The vaccines need to be given within 6 months of your entering Belize.
I drove here from the states and the Mexican officials barely noticed I had 2 cats in a carrier.....let alone checking for documents.
At the border of Belize and Chetumal, They did check the documents....I paid a small fee and they stamped the document....but they never asked to see the cats. Go figure!
I don't know if the procedure is the same if your coming by air. I only know my own personal experience.
The procedure is very different for farm animals and horses.
Hope this helps