The simple answer to this question is no. You don’t need a formal signed paper contract between you and your client. However, you do need a written agreement in some form that details the job, the price, and the date the job is going to be completed by. The written agreement can be via email; although signed contracts clearly are the most robust.
The reason for having a written agreement is that if there is any dispute about the work that you have done we can help defend you. Or if a customer doesn’t pay you for work you have done, without a written agreement it will be very difficult to help chase your invoice.
As they say: “A verbal contract is only worth the paper it’s written on.”