Putting it on your phone is great but also make a hard copy jic. If the phone gets lost or batteries run down the copy will get you through the lines. We brought ours with us and as previously suggested, we left them in the safe with our passports.
"Always keep your words soft and sweet, just in case you have to eat them."
That is what we have. If you have like an all in one printer. Put your vax card and copy at 90%, copy both sides, get the laminate pack at Walmart (office supply section) pack of 5 for like $3-$4. Then it is about the size of the typical credit card and you can just put it in with your credit cards and keep the original in a safe place.