You should be able to send a letter that serves as a contract, and include an invoice to pay to start the contract.

Yeah, but including a letter clearly explaining a service contract isn't going to fool many accountants. And if it's not clearly stated in the contact letter what exactly the invoice is for, it's still fraud.

If you're going to commit fraud, just don't do it by mail/wire. Federal prosecutors have a 95% conviction rate, and the maximum sentence for mail fraud is ten years longer than bank robbery.