The cards need to be both protected and available to show when needed. You can carry the original paper card with you, plastic coated perhaps. But lots of people stash it away and show a digital copy.

(Image credit: Rogelio V. Solis/AP)

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Source: https://www.npr.org/sections/coronavirus-live-updates/2021/08/10/1026243726/how-to-keep-vaccination-confirmation-cards-with-you