Is it possible to fake an ankle sprain and get crutches?

Committing Fraud. It is possible, but if you are not paying for them then that would be an illegal and unethical act. If you want crutches you can rent them easily from most pharmacies. And, you may be able to borrow them from your local fire station. Many fire stations offer them as a community service.
Sure! But why stop there? Why not go for a wheelchair and a handicapped accessible van with lift and handicapped parking sticker. The sky is the limit. What's to stop you? Oh yes, your conscience and the law... Walk it off.