I have a SharePoint 2013 add-in that uses the SP REST API to add an item to a library on the host web. I don't want to grant app users edit-level permissions to the library since the app will be in charge of adding library items, not the user. I have already set the app's permission level in the app manifest to full-control, however the app's Post call returns an error when a user with read permissions attempts to use the app. Is there anyway for a user with read permissions to use an app that Posts to a host library?
Just use Add-in-only policy. In Visual Studio, open app manifest in design mode and there is a checkbox in upper left corner on the permissions tab. Add-ins-only policy enables Add-ins to use separate access rights in given scope. It should do the job for you.
You cannot use a Rest Post action on host web with users that have read permission. Check this link and read User+add-in policy : Get an overview of add-in authorization policies types