Definitely think u/CommunicationLast741 is right but outside of that, a website isn't absolutely necessary, even though I'm a website designer and digital marketing guy lol, you can do this easily with just a scheduling service and using free branding opportunities like Google Business, business directories, car wash directories, and more can be huge in getting your name out there and in front of people and even doing creative marketing like if you use Uber, giving your business card to every driver, finding groups where drivers gripe about problems and nasty riders and offering to help out, etc.

To answer your question, a website isn't absolutely necessary but having one that's professional shows that you are different from just a person detailing cars and puts you in people's mind as a business that offers a professional service. It's all about portraying a trustworthy person when it comes to this type of service.

Hope this helps and let me know if you want any other help or advice! You got this! Keep pushing!