You can certainly add your personal contact info in your profile bio, that's more or less why we have it there.
However, you can't do that at the bottom your answers. Our policy here is to keep the answers as clear as possible of any personal/self-promoting information because, well, the purpose of Socratic is to help students learn, not to give people a platform to promote themselves on.
You are right, everything is free here, which is why it's important to keep things as spam-free as possible, and we feel like self-promoting in every answer you post is a clear-cut case of spamming.
Now, you can create an account with the same username as your website, as long as you keep things reasonable, i.e. don't go full http://youcanfindmehereforprivatetutoring.com as the username [I made that one up for illustration purposes, haha]
Here's an example of how you could use your username and your profile bio to promote yourself and/or your site
Long story short, you can definitely use your bio section to post your contact info if you want, but don't tag that along in your answers.