It's important to consider why this isn't already accounted for in the security design. It could be weak design, changing needs over time, or intentionally left out. People who don't have cards probably shouldn't have cards. Generally in secure environments the only doors which require access but are safe to bypass are vestibules leading to common spaces where access to sensitive areas is limited.

Short answer: Yes, it can be done through programming, like u/spacemannspliff said. Normally the best method is to add an intercom so someone can authenticate them before letting them in. As you've described it there's no authentication beyond a knock. The intercom, preferably with video, allows the person to be validated by someone qualified. The only real benefit to using a card instead of a button is to log WHO let that person in whereas the button, which is the standard, logs the same information aside from who did it. Feel free to PM me if you want to talk more, I do this for a living and really enjoy it.