Applications to be a contestant on a game show are generally available on that particular show's website or on the production company's website. Applying is generally only the first step, followed by some sort of audition or casting call.
There are some game shows that don't have an application process, such as "The Price is Right." This show generally picks contestants from ticket holders to the show's taping. The show does do traveling auditions, but the majority of contestants are chosen on the spot.
"Jeopardy" requires successful completion of an online trivia test before casting. "Wipeout," a physically challenging game show, requires the navigation of an obstacle course after initial application to make it onto the show.