Actually, I hear this question a lot. For both the environment and your sanity, streamlining the finance process to minimize paperwork is a must. And yet, there is a definite need for documentation and accountability. The balance can be achieved by careful consideration, adjustment, and evaluation.
First, look at what you can do online. Google docs is a program that lets you access shared excel files that multiple users can contribute to. Online forms are available, which can be filled out, emailed, and stored as a file. This is usually an easier, more convenient option and still allows for accountability. In addition, look into programs that will help make documentation easier. Anything that automates the process or lets you enter data and generate reports can mean fewer headaches and less paperwork. Using corporate cards to pay for expenses can mean all costs are listed in the card statement, which lets you see exactly where and how much is being spent.
Whatever you can move onto the computer into an efficient system will cut down on both actual paper and paperwork.