When I go to collections, I don't have the ability to reclaim any exotic weapons or armor. Happens on all exotics, not just those in inventory. Legendary weapons can be reclaimed just fine.
As you can see in the screenshot, the cost to reclaim the item is visible, just no reclaim option. Others in my clan do not have this issue.
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That's because you're looking in the badge section. You need the exotic section which is on the left of the main collections page.