Unfortunately, twitch prime rewards are impossible to be claimed if there are too many items awaiting retrieval. The NPC system does not seem to support high volume of items.
I should have more than 100 twitch prime items to be claimed, but the NPC just bugs out as seen on the video link. There's a clear "item hover" indicator which spams like crazy when hovering the empty space where the items should be.
Require Bungie support to be able to claim these. (STEAM)
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Swat
Bungie fan! Here since 2008 - old
Bungie is aware. It will be patched at some point in the future.