I recently completed all the lightfall campaign quests on legendary. I see the achievement in the menus. However, after speaking to Zavala, Ikora, and finally Nimbus I didn’t get any of the exotic rewards other than an emblem to begin a quest for a “heavy” weapon exotic.
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Hi there. Your Titan's history shows you did the missions "First Contact" and "Under Siege" on Classic difficulty. You got the legendary campaign emblem because you did those missions on Legendary with your other characters. To get the full rewards, including exotic armor, you need to do all of the missions on Legendary difficulty with the same character.