Can I shoot an elk for my disabled mother and tag it with her tag?

No, that would be entirely against the law. Fish and Game tries to accommodate folks who are not able-bodied with special license fees for the disabled and even with shoot-from-a-vehicle permits in some special cases, but shooting a game animal for someone else has always been unlawful. "Party hunting" with shooting until all the tags in the group are filled is one of the practices the law is intended to halt, and tickets are regularly issued when it is detected. Essentially, there would be no place to stop if hunters could legally shoot for someone else.