Maybe it's a Midwest thing, but I still believe that shaking someone's hand on a deal is as good as a contract. Even if that handshake is over the phone on in an e-mail. For that reason, I don't require a contract for most of my Summer Reading Program Library Magic Shows.
If a library doesn't know me and would feel more comfortable with a contract, I am more than happy to draw something up and either mail it or e-mail it to them. The same holds true for invoices. I know that many libraries need the paperwork for record-keeping.
I think it's also wise to have something in writing if I'm doing several shows for a library system or if the payment amount is unusually high.
The contracts that I draw up just simply state what our agreement is...I will do a show at your library on at such and such a time on such and such a date and I will be paid X amount of dollars for the performance. Don't worry, I don't have any riders in my contract that state you must provide me with 2 bottles of a certain brand of water chilled to 36 degrees and a bowl full of chocolate covered grapes. I prefer being easy to work with. In fact, when you look back at your Summer Reading Program events, I want you to think of me as one of the easiest, least-stressful events you had.
So, if you need a contract or invoice, let me know. Otherwise, a long-distance handshake works well for me.