"Loan request" is a possibility.

You "borrow" an item from somebody, but you "request a loan" of an item. 

Although "loan" is often used when talking about money, it is not restricted to that context. 

If you want to borrow a book and your local library does not have it, you put in an [Interlibrary loan request][1].

  [1]: http://www.illawarra.tafensw.edu.au/support-for-students/library-and-information-services/campus-libraries/contact-your-library/interlibrary-loan-request