No - today a Zoom App can’t capture the user’s mic for arbitrary “voice memo” recording. Zoom has confirmed the Zoom client proactively blocks navigator.mediaDevices.getUserMedia() in Zoom Apps and that Zoom Apps don’t currently provide access to media devices.
Because of that, standard browser recording flows that rely on getUserMedia() won’t work inside the in-client embedded browser. The practical workaround is to open your existing recorder page in the system browser via zoomSdk.openUrl (note the domain whitelisting requirement called out in that same doc section).
There isn’t a Marketplace permission you can toggle to enable mic capture inside the Zoom App; it’s a client-side privacy/security restriction. If your requirements ever shift to meeting audio/video, Zoom’s supported paths are Meeting SDK raw data or Realtime Media Streams (RTMS) (both are for meeting media, not standalone mic memos).