I've had 2 unpaid internships in while I was at a university, both unpaid, no such thing needed for work authorization. The stringent background check/requirement only comes with the type of field, obviously you can't expect the same process for an insurance sales internship as opposed to a nuclear physist internship.

Also note that I receieved college credit for both. In most cases, you don't even have to point out anything, since they won't make you fill out an I-9 or anything like that.