"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.

"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.

"Loan request" is a possibility.

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

"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.