This question may have been asked in the past, but not sure if it was in context of SP 2013 or SP online. We have a need for users to be able to add items to a custom list using a form (list form), but not being able to view the list itself. Only a select group of staff should be able to view the list.

You can set users with contribute permission to the list and set Item-level Permissions so that users can only read items that they have created.
List Settings > Advanced Settings > Read items that were created by the user 

The other option is to programmatically impersonate the user to submit data in to the list without granting access to them to

Edit the list view web part page and set Target Audience Property for the web part. This would not affect any permissions associated with the particular list