Very interesting to see another emergency use case !
For your problem with the app to locate nearest landing zone : maybe the current apps could help with that (osmand or magic earth) as they pre-load POI information. I don’t know which tag is used for these landing zone, but for a simple helipad (aeroway=helipad), i can find them with magic earth or osmand search (and it list them with distance AND show them on the map). If the tag used for these landing pad is not loaded in the database of the app, you could always ask them to do it. At least that would allow to do it without developing another app ! :-)
Then for better usability, i don’t know if that possible to create a shortcut button to automatically launch this search instead of manually entering it (i’m not really used to iOS or Android capabilities).