I would like to learn if this use of 'would' is possible.
A would-be dancer begins training at the age of 7 or 8.
Yes. It is a very common expression, and means, in this example, someone who very much wants to be a professional dancer must start their training very early.
A spin-off from that expression is someone described as a 'wanna-be', which means somebody who craves celebrity for the sake of it. They may have little or no talent, and become 'famous for being famous'.