I am not very sure if any such searchable database or a website is even possible for anyone to create (including Google itself), given the current size of Android apps, the indie app development ecosystem of Android and plethora of Android devices on offer, which varies from country to country and more.
I guess the simplest approach is to take it case-by-case for each app.
If you find a new/unknown app, just do a quick Google Search using app's Name.
Most likely that you will find a link to Either Google Play store (if that app is hosted on Google Play) or some other Link from where you can download the APK for the app. Developers tend to provide enough information (such as app permissions) about the app so that user can see and install apps through these links.
Having said that, I found few useful links/sites which can be visited for app info.
Of course, these links/sites have their limitations and they do not have EVERY Single app which is ever developed for Android.