In this phrase, 用 is a verb which can be used both transitively and intransitively (i.e. it can be used either with or without an object). 一下兒 is an adverbial of time indicating "a short while, a little while". If an object is to be used for 用 in the above phrase, it should be placed after 一下兒:
我用一下兒你的書﹐好嗎﹖
Is it all right for me to use your book for a short while?
他們用一下兒你爸爸的車。
They want to use your dad's car for a little while.