When speaking about these kinds programmes the phrase that is often used is that the person will get a 'record deal.' It sounds a bit more impressive.
The finalist (do you mean 'winner?' Seems a bit weird that you only have to reach the final to get the prize but anyway) will get their own record deal/sign a contract with a record label, taking the first step in their music career.
Your sentence is almost OK, it should be: The finalist will have the opportunity to record their own CD with a popular company.