How can I browse or search the iOS app store from my desktop?
Apple have very quietly released iTunes 12.6.3 - compatible with iOS 11 &amp; still containing the App Store.  

Apple KB: Deploy apps in a business environment with iTunes

You need the direct link, it apparently does not appear as an update.
Mac link.
Windows 32 bit link.
Windows 64-bit Link.

If you've already been using 12.7 you may have luck importing your 12.7 lib  to a clean 12.6.3 Library, as outlined on MacRumors, but I had no joy.  

I ended up restoring a backup of my Library [just the catalog not all the tunes] from a time before i updated to 12.7.