How can I check what permissions an app store app will request before installing it?
The iOS App Store does not display the permissions the app will request.  You can view certain info about an app prior to downloading:


Seller
Size
Category
Compatibility
Languages
Age Rating
In-App Purchases
Copyright
Developer Website (link)
Privacy Policy (link)


Unless the developer indicates the required permissions in their app description, Privacy Policy or Website, you will not know until the app requests that permission.