By "each time you collect x badges", he means "each time Agent O receives x number of badges". The badges are supposed to be turned in for this quest. The reward is just exp most of the time and a pair of gloves every 10th time you turn it in. I know this because I have done it 20+times.

Thanks Kiji, and thanks boggart for the perfect explanation! Agent O doesn't hand badges out, he collects them. Stay in a Mini Dungeon for about 30 minutes and an Agent Box appears; open it for a Special Agent Badge and bring the badge to Agent O. Each hand-in pays EXP, every 10th one adds a pair of gloves. That matches the original game, so nothing to fix here.