No. Your cat is not a product, and Amazon has policies against selling sentient beings. But you can sell cat-themed knick-knacks and make a nice little profit.
Step 1: Create a product with an absurdly long name. Step 2: Post a picture of it on your cat. Step 3: Wait for the algorithm to magically decide your product is the best thing since sliced bread. If that doesn't work, hire a pack of robotic reviewers to tell you how great it is.
Sorry, but Amazon Prime is only for products that don't make you cry. You can contact our Air-Filled Products Department for further assistance.