Start with almost any good-quality sound recording app -- it does not need to be bioacoustics related, any good "sound recorder". There are many apps, and the best choice keeps changing, so I make no recommendations by name. Instead, pick an app for which:
- you can record uncompressed audio (e.g. WAV or FLAC) -- check this, since most "speech diary" apps only record poor-quality low-filesize audio formats
- you can access the recorded files
- allows you to control the gain
- ideally visualises the audio in real-time: a gain/amplitude level and/or a spectrogram.
Since your question is about exploring the possibilities, I do not recommend using a task-specific app such as BirdNet or Warblr: both are bird sound recognisers, but they're not designed for general-purpose recording nor for accessing the files afterwards.
You should then ask a small group of users to make recordings with their own phone, in their own chosen location, and to send the files to you. Pretty quickly, with a small group of users (less than 10), you will have a first idea of the diversity of recording quality, noise levels, user error, for your goal.
When you have collected the audio files, you can use analysis software to determine what's possible (e.g. automated recognition), before going further and trying to connect an app to some more involved system.