New Jersey residents can easily access their vaccination information and records through the Docket app. This will likely be the most convenient method of showing proof-of-vaccination for most individuals, and is available to Apple and Android store users. More information is here: https://covid19.nj.gov/pages/vaxrecords
For those using a paper card and in need of a replacement, information on that is available here: https://covid19.nj.gov/faqs/nj-information/slowing-the-spread/how-do-i-get-a-new-copy-of-my-covid-19-vaccination-card-is-there-an-app-or-a-digital-copy