As with all the best questions, I'll answer this one two months later. Don't worry, I've definitely been spending all this time giving it lots of thought.
Hagane became a Huntsman as sort of an honorable thing, out of disgrace for his parents who are irresponsible and cowardly.
Mint became a Huntress due to her interest in exploring and how relatively confined she was at home.
Persimmon became a Huntsman for glory. He wanted people to think he's a cool dude.
Carnation became a Huntress to protect people. It was her way of giving back to the community.
Lissa just always knew she was going to become a Huntress. It runs in the family and seemed like enough fun for her.
Patchouli read about the Huntsmen and even did a play on them, which eventually got her hooked.
Diamante has been experiencing the average person's work for ages thanks to his noodle shop and still does. He didn't like it. Fighting seemed more interesting.