Email / Form authentication

Don't know what this is but I'm sure somebody will.
I'd like to put a new booking form on my site, now that bit is easy, but I'd like to authenticate in some way with a signature.

Currently the customer submits a HTML form where the input is sent to a PHP file, I receive the details via email and at the same time, the customer receives a ‘Thank You’ email.

Now what I would like to do is somewhere along the lines of:
Customer submits form.
I receive form details via email
Customer receives confirmation to the email address submitted.

But then I’d like to add something like a link, whereby the customer has to reply to the link to authenticate that he/she was the person who sent the original email.

This would then act as an electronic signature.

Or do any of you have other / better ideas?