"Borrow request" is not correct, but in the context of a game people will understand what you mean.
"Loan request" is an alternative that is shorter than "borrowing request" and more common books.google.com/ngrams...
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.