Search API Views Indexed Node Filter by Author Role
You need to index the author's roles so that Search API can query that field.  

Visit the fields configuration page for your node index (probably admin/config/search/search_api/index/default_node_index/fields if you're using the default node index).  At the bottom of the list of fields you'll see a fieldgroup called "add related fields".  Expand that, choose author, click "add fields".  

When the page reloads your big list of fields will now contain a bunch of new Author-related fields including Author » User roles.  Check the box next to that one, choose how you want it indexed (integer or string are probably the only ones you'd care to use for this purpose), save your changes, and re-index your content.

I've found that with these sorts of changes I need to clear my cache twice in order for the proper data to be registered with Views and for the appropriate filters to appear in my Views.