It is currently not possible to fetch a user’s owned bundles with the open cloud API using an API key or OAuth2 token. The /cloud/v2/users/{user_id}/inventory-items open cloud endpoint, which already has support for other item types like badges and gamepasses, does not allow for the inclusion of bundles.
Also, since some Roblox-published bundles are limited [1] [2], bundle support on this endpoint should ideally support the onlyCollectibles filter