“Through the Costco Mobile App, you can access a digital version of your membership card. Costco’s app will also allow you to renew your membership and view your 2% reward balance for executive members on your phone.”
The only caveat is from the FAQ “Note: Currently the Digital Membership Card is not accepted at Costco Gas Stations.”
That means you can’t yet take your Costco card out of your wallet or purse if you buy your gas there. Note that in some states a membership is not required to access Costco’s Gas station.